541.1 

W27t 

1914 


E W.  Washburn 


"FVinci  pies 
c.hemis't'ry 


of 


JOHN  W.  DAVIS 
INTRODUCTION 

TO  THE 

PRINCIPLES 

OF 

PHYSICAL  CHEMISTRY 


PART  I 


BY 

EDWARD  W.  WASHBURN 

PROFESSOR  OF  PHYSICAL  CHEMISTRY  IN  THE 
UNIVERSITY  OF  ILLINOIS 


Preliminary  Edition 


McGRAW-HILL  BOOK  COMPANY,  Inc. 
239  WEST  39TH  STREET,  NEW  YORK 

6 BOUVERIE  STREET,  LONDON,  E.  C. 

1914 


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PRINCIP^§  OF 
PHYSICAUeHEMISTRY 


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iiii2 


j 1 


INTRODUCTION 


TO  THE 


Gy 


PRINCIPLES 


OF 


PHYSICAL  CHEMISTRY 


PART  I 


BYor 

EDWARD  W.  WASHBURN 

PROFESSOR  OF  PHY8I(A  CHEMISTRY  IN  THE 
UNIVERSITt^F  ILLINOIS 


Preliminary  Edition 


McGRAW-HILL  BOOK  COMPANY,  Inc. 
239  WEST  39TH  STREET,  NEW  YORK 
6 BOUVERIE  STREET,  LONDON,  E.  C. 

1914 


Copyright,  1914,  by  the 
McGraw-Hill  Book  Company,  Inc. 


>.  hill 


'^^lL  HEMQTE  storage 

CONTENTS 

CHAPTER  ’ PAGE 

I.  The  Structure  of  Matter  and  the  Composition  of  Sub- 

stances 1 

II.  The  Gaseous  State  of  Aggregation 19 

III.  The  Liquid  State  of  Aggregation 40 

IV.  Liquid-Gas  Systems 53 

V.  The  Crystalline  State  of  Aggregation 65 

VI.  Crystal-Gas  Systems 71 

VII.  Crystal-Liquid  Systems 73 

VIII.  Relations  between  Physical  Properties  and  Chemical 

Constitution 78 

IX.  The  Brownian  Movement  and  Molecular  Magnitudes.  . 85 

X.  Some  Principles  Relating  to  Energy 93 

XI.  Solutions  I:  Definition  of  Terms  and  Classification  of 

Solutions 114 

XII.  Solutions  II:  The  Collegative  Properties  of  Solutions 

and  the  Thermodynamic  Relations  Which  Connect 
Them 122 

XIII.  Solutions  III:  Thermodynamic  Environment.  Ideal 

Solutions  and  Dilute  Solutions 134 

XIV.  Solutions  IV : The  Laws  of  Solutions  of  Constant  Ther- 

modynamic Environment 143 


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PRINCIPLES  OF 
PHYSICAL  CHEMISTRY 


CHAPTER  I 

THE  STRUCTURE  OF  MATTER  AND  THE  COMPOSITION 
OF  SUBSTANCES 

1.  The  Rise  of  the  Atomic  Theory.2 — -The  problem  of  the  consti- 
tution of  material  bodies  is  one  which  has  interested  philosophers 
since  the  dawn  of  history.  Some  time  prior  to  500  B.  C.,  the 
Hindu  philosopher  Kanada  held  the  theory  that  material  bodies 
are  not  continuous  but  are  made  up  of  very  small  indivisible 
particles,  which  are  in  a constant  state  of  motion,  flying  about  in 
all  directions.  This  theory,  known  as  the  atomic  theory,  was  also 
advanced  about  500  B.  C.  by  Leucippus,  the  founder  of  the  Greek 
atomistic  school,  and  was  further  developed  by  his  pupil  Democ- 
ritus to  whose  writings  we  owe  all  of  our  knowledge  of  the  begin- 
nings of  the  atomistic  philosophy  of  the  Greeks.  These  ancient 
atomic  theories  were,  however,  entirely  the  result  of  purely  meta- 
physical speculation  without  any  experimental  basis  and  until  the 
birth  of  experimental  science  no  further  progress  in  this  direction 
was  possible. 

The  first  noteworthy  attempt  to  interpret  the  known  behavior 
of  material  bodies  in  terms  of  the  atomic  theory  was  made  in 
1743  by  Lomonossoff,®  a Russian  physical  chemist,  but  his  work 
unfortunately  remained  unknown  outside  of  Russia  until  1904. 

a Michael  Vassillevitch  Lomonossoff  (1711-1765),  born  of  peasant  parents 
in  a little  village  near  Archangel,  became  professor  of  chemistry  in  the 
Academy  of  Sciences  in  St.  Petersburg  and  in  1748  built  the  first  chemical 
laboratory  for  instruction  and  research.  His  publications  stamp  him  as  one 
of  the  greatest  of  physical  chemists.  His  ideas  of  elements,  molecules, 
atoms,  heat,  and  light  were  essentially  the  same  as  those  held  today  and  in 
many  ways  he  was  so  much  ahead  of  his  time  that  his  work  was  ridiculed 
and  forgotten  until  resurrected  by  Menschutkin  in  1904  (See  Alexander 
Smith,  Jour.  Amer.  Chem.  Soc.,  34,  109  (1912)). 

1 


2 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


Meanwhile  in  1803,  John  Dalton,®  in  England,  had  given  to  the 
world  the  modern  atomic  theory,  which  has  played  such  an  im- 
portant role  in  the  development  of  the  science  of  chemistry. 
Although  this  theory  proved  of  the  greatest  assistance  in  the  inter- 
pretation and  correlation  of  the  known  facts  of  chemistry  as  well 
as  in  pointing  the  way  toward  new  discoveries,  it  was  nevertheless 
merely  a theory,  for  however  useful  atoms  and  molecules,  as 
concepts,  might  be  in  interpreting  the  data  of  science,  they  have, 
until  very  recently,  been  regarded  merely  as  convenient  hy- 
potheses, not  as  actualities. 

During  the  last  decade,  however,  a flood  of  new  and  more 
refined  methods  of  investigation  has  thrown  a powerful  light  upon 
the  old  question  of  the  structure  of  matter.  This  new  evidence 
is,  moreover,  of  such  a striking  and  convincing  character  and 
comes  from  such  a great  variety  of  different  sources  that  chemists 
and  physicists  of  the  present  day  may  now  accept  the  atomic  and 
molecular  structure  of  matter  as  a fact  established  beyond  the 
possibility  of  reasonable  doubt.  Some  of  the  more  important 
pieces  of  evidence  which  have  led  to  this  result  will  be  presented 
during  the  development  of  our  subject. 

2.  The  Structure  of  Matter. — Our  present  qualitative  knowl- 
edge concerning  the  structure  of  matter  and  the  composition  of 
chemical  substances  may  be  briefly  summed  up  as  follows: 

(a)  Atoms. — Every  elementary  substance  is  made  up  of 
exceedingly  small  particles  called  atoms  which  are  all  alike  and 
which  cannot  be  further  subdivided  or  broken  up  by  chemical 
processes.  It  will  be  noted  that  this  statement  is  virtually  a 
definition  of  the  term  elementary  substance  and  a limitation  of 
the  term  chemical  process.  There  are  as  many  different  kinds 
of  atoms  as  there  are  chemical  elements. 

(b)  Molecules. — Two  or  more  atoms,  either  of  the  same  kind 
or  of  different  kinds,  are,  in  the  case  of  most  elements,  capable 
of  uniting  with  one  another  in  a definite  manner  to  form  a higher 
order  of  distinct  particles  called  molecules.  If  the  molecules  of 
which  any  given  material  is  composed  are  all  exactly  alike,  the 

a John  Dalton  (1766-1844),  Tutor  in  Mathematics  and  Natural  Phil- 
osophy in  the  New  College,  Manchester.  His  New  System  of  Chemical 
Philosophy  appeared  in  1808. 


Sec.  2] 


THE  STRUCTURE  OF  MATTER 


3 


material  is  a pure  substance.  If  they  are  not  all  alike,  the 
material  is  a mixture. 

(c)  Elementary  Substances. — If  the  atoms  which  compose 
the  molecules  of  any  pure  substance  are  all  of  the  same  kind,  the 
substance  is,  as  already  stated,  an  elementary  substance.  It  is 
evident,  however,  that  different  elementary  substances  may  be 
formed  from  the  atoms  of  the  same  element  according  as  the  mole- 
cules which  these  atoms  form  are  composed  of  one  (monatomic 
molecules),  two  (diatomic  molecules),  three  (triatomic  molecules), 
or  more  (polyatomic  molecules)  atoms  per  molecule  and  even  in 
the  case  of  molecules  composed  of  the  same  number  of  atoms  all  of 
the  same  kind,  substances  of  different  physical  properties  may  be 
produced,  if  the  atoms  within  the  molecule  are  differently  art 
ranged  with  respect  to  one  another.  Hence  the  same  element 
may  exist  as  various  different  (allotropic)  elementary  substances. 
Thus  there  are  several  perfegtly  distinct  substances  known,  each 
of  which  when  allowed  to  unite  with  oxygen  will  produce  2 grams 
of  pure  sulphur  dioxide  for  each  gram  of  the  substance  taken. 
This  proves  that  the  atoms  of  all  of  these  substances  are  sulphur 
atoms  and  the  substances  are  all  known  as  different  forms  of  the 
element  sulphur.  Their  different  physical  properties  are  due  to 
differences  in  the  internal  structure  or  the  arrangement  of  the 
molecules. 

It  may  also  happen  that  a given  material  might  display  nearly 
all  of  the  chemical  and  physical  properties  characteristic  of  an 
element  and  still  be  composed  of  more  than  one  kind  of  atoms. 
This  situation  has  arisen  recently  in  the  case  of  the  element  lead. 
Richards®  has  found8  that  the  atomic  weight  of  the  lead  ob- 
tained from  radioactive  minerals  is  quite  appreciably  different 
(0.36  per  cent.)  from  that  of  ordinary  lead.  The  ultra  violet 
spectra  of  the  two  materials  were  nevertheless  found  to  be  en- 
tirely identical,  line  for  line,  and  in  their  chemical  behavior  the 
two  “leads”  were  indistinguishable  from  each  other.  No  sepa- 
ration of  either  material  into  two  or  more  different  substances 
could  be  effected  and  when  once  mixed  together  not  the  slightest 
separation  of  one  from  the  other  could  be  effected  by  any  physical 
or  chemical  means  tried.  The  only  way  in  which  the  two 

° Theodore  W.  Richards  (1868-  ),  Professor  of  Chemistry  in  Harvard 

University. 


4 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


materials  could  be  distinguished  from  each  other  at  all  was 
through  their  different  atomic  weights  and  through  the  fact  that 
one  material  was  radioactive  (I,  2f),  thus  showing  the  presence 
of  atoms  in  an  unstable  condition.  It  may  be  mentioned  here 
that  recent  theories6  advanced  by  Soddy®  and  by  Fajans6  predict 
the  existence  of  groups  of  elements  which  are  chemically  non- 
separable  from  one  another  and  which  differ  only  in  the  different 
atomic  weights  of  the  members  of  the  group  and  in  the  different 
degrees  of  stability  of  their  atoms.  Several  such  groups  have 
been  studied  by  Socldy.  It  has  been  proposed  to  call  the  mem- 
bers of  such  a group  nonseparables  or  isotopes.  They  will  receive 
a more  detailed  treatment  in  a later  chapter. 

(d)  Compound  Substances. — If  the  atoms  which  compose  the 
molecules  of  a pure  chemical  substance  are  not  all  of  the  same 
kind,  the  substance  is  a compound  substance.  Just  as  in  the 
case  of  elementary  substances  there  may  be  several  different 
compound  substances,  all  composed  of  molecules  having  the  same 
atomic  composition  and  owing  their  different  properties  to  the 
different  ways  in  which  the  atoms  composing  the  molecule  are 
arranged  with  respect  to  one  another.  Thus  two  different  sub- 
stances are  known  whose  molecules  are  each  composed  of  two 
atoms  of  carbon,  six  atoms  of  hydrogen,  and  one  atom  of  oxygen, 
that  is,  the  molecules  of  both  substances  have  the  composition 
represented  by  the  empirical  formula,  C2H60.  The  molecules 
of  one  of  these  substances  have  the  structure  which  is  represented 
by  the  formula 

H H 

H-C-C-O-H 
H H 

This  substance  is  ordinary  ethyl  alcohol. 

The  structure  of  the  molecules  of  the  second  substance  is 
represented  by  the  formula, 

H H 

H-C-O-C-H 
H H 

° Frederick  Soddy,  F.  R.  S.  (1877-  ).  Formerly  Lecturer  in  Physical 

Chemistry  and  Radioactivity,  at  the  University  of  Glasgow.  Since  1914 
Professor  of  Chemistry  at  the  University  of  Aberdeen. 

6 Kasimir  Fajans,  Investigator  in  the  Laboratory  of  Physical  Chemistry 
of  the  Karlsruhe  Institute  of  Technology. 


Sec.  2] 


THE  STRUCTURE  OF  MATTER 


5 


and  this  substance  is  called  methyl  ether.  Compound  substances 
whose  molecules  are  identical  in  composition  but  different  in 
structure  are  called  isomeric  substances  or  isomers. 

It  is  obvious  that  the  more  nearly  two  molecular  species  resem- 
ble each  other  both  in  composition  and  in  structure,  the  more 
closely  will  the  two  substances  composed  of  these  molecules 
resemble  each  other  in  all  of  their  physical  and  chemical  properties. 
In  some  cases  this  resemblance  is  exceedingly  close.  Thus  there 
are  two  distinct  substances  known,  both  of  which  are  called  amyl 
alcohol  and  both  of  whose  molecules  have  the  structure  repre- 
sented by  the  formula 


CH, 


H 


\c/ 

/c\ 


C,H; 


CH,OH 


In  nearly  all  of  their  physical  and  chemical  properties  the  two 
substances  are  identical.  They  both  have  the  same  melting  point, 
the  same  boiling  point,  the  same  heat  of  combustion  and  the  same 
solubility  in  water.  The  chief  difference  between  them  lies  in 
their  behavior  toward  polarized  light  and  this  behavior  serves 
as  a means  of  distinguishing  one  from  the  other.  If  a beam  of 
polarized  light  is  allowed  to  pass  through  a layer  of  the  substance 
in  the  liquid  or  the  gaseous  state,  the  plane  of  polarization  is 
rotated  to  the  left  by  one  substance  while  the  other  rotates  it 
to  the  right  to  an  exactly  equal  degree.  The  first  substance  is 
called  laevo-amyl  alcohol  and  the  second  substance  dextro-amyl 
alcohol  for  this  reason.  The  difference  in  molecular  structure 
which  is  the  cause  of  this  behavior  is,  according  to  the  theory  of 
van’t  Hoff  a-LeBel,6  simply  a difference  in  the  order  in  which  the 
four  different  groups  are  arranged  in  space  about  the  central 


“Jacobus  Henricus  van’t  Hoff  (1852-1911),  Professor  of  Chemistry  in 
the  University  of  Amsterdam  (1877-1896)  and  in  the  University  of  Berlin 
(1896-1911).  He  met  Le  Bel  while  studying  with  Wiirtz  in  the  Ecole  de 
Medecine  in  Paris  in  1874  and  his  paper  on  the  asymmetry  of  the  carbon 
atom  appeared  in  September  of  that  year.  Van’t  Hoff  is  especially  re- 
nowned for  his  contributions  to  the  modern  theory  of  solutions. 

6 Joseph  Achille  Le  Bel,  F.  R.  S.,  a French  chemist  residing  in  Paris. 


6 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


carbon  atom,  the  arrangements  in  the  two  cases  being  such  that 
one  molecule  has  the  same  arrangement  as  the  mirror  image  of 
the  other.  Such  isomers  as  these  are  called  optical  isomers  and 
occur  whenever  a molecule  is  made  up  of  four  different  atoms  or 
groups  all  attached  to  the  same  central  atom.  The  central  atom 
under  these  conditions  is  said  to  be  asymmetric. 

(e)  Electrons. — We  have  seen  that  the  molecule  of  a substance 
is  an  individual  composed  of  one  or  more  atoms  and  its  structure 
may  be  a very  complex  one,  if  it  happens  to  be  made  up  of  a large 
number  of  different  atoms.  Modern  research  has  demonstrated 
that  the  structure  of  the  atom  is  likewise  very  complex.  It  has 
been  shown  that  every  atom  contains  a considerable  number  of  a 
third  and  lower  order  of  distinct  particles  called  electrons  or 
corpuscles.  These  particles  seem  to  be  nothing  but  free  charges 
of  negative  electricity.  They  are  all  exactly  alike,  regardless  of 
what  atom  they  may  come  from,  and  each  constitutes  the  small- 
est quantity  of  electricity  capable  of  existence.  In  other  words 
electricity,  like  matter,  is  atomic  in  structure  and  the  electron  is 
the  “atom”  of  electricity.  The  mass  (in  the  usual  sense  of  this 
term)  of  an  electron  is  not  known,  but  some  recent  experiments 
by  Tolman®  indicate  that  it  must  be  less,9  and  probably  consid- 
erably less,  than  tto  of  the  mass  of  the  hydrogen  atom,  the 
lightest  of  all  the  atoms.  The  electromagnetic  mass  of  an  electron 
is  known  to  be  only  tsVo  as  large  as  the  mass  of  the  hydrogen 
atom.  The  electron  is  thus  the  smallest  particle  of  which  we 
have  any  knowledge. 

(f)  Radioactivity.3 — In  addition  to  these  electrons,  which  seem 
to  be  part  of  the  composition  of  every  atom,  there  are  also  other 
components  about  which  very  little  is  known  with  certainty  at 
present.  All  of  these  components  are  held  together  by  the  intra- 
atomic  forces  to  form  the  complex  individual  system  which  we 
call  the  atom.  The  atoms  of  most  elements  are  exceedingly 
stable  and  practically  indestructible  systems,  but  in  some 
instances,  especially  in  the  case  of  the  larger  and  hence  probably 
more  complex  atoms  such  as  those  of  the  elements  radium, 
uranium  and  thorium,  the  atomic  systems,  from  time  to  time, 
reach  a condition  of  instability  which. results  in  a complete  break- 

“ Richard  Chase  Tolman  (1881-  ).  Assistant  Professor  of  Chemistry- 

in  the  University  of  California. 


Sec.  2] 


THE  STRUCTURE  OF  MATTER 


7 


ing  up  of  the  atom.  The  various  components  of  the  original 
atom  then  rearrange  themselves  into  new  atomic  systems,  that 
is,  new  elements  are  produced.  This  atomic  disintegration  may 
be  accompanied  by  the  violent  expulsion  of  a stream  of  electrons 
known  as  /3-rays  or  it  may  be  accompanied  by  the  expulsion  of  a 
stream  of  helium  atoms  each  carrying  two  unit  charges  of 
positive  electricity  and  known  as  a-particles.  Such  disintegra- 
tions are  known  as  radioactive  changes  and  are  evidently  quite 
distinct  from  chemical  reactions,  which  involve  the  interaction 
of  two  or  more  atoms  or  molecules.  All  radioactive  disintegra- 
tions, thus  far  known,  are  perfectly  spontaneous  and  appear  to  be 
unaffected  by  any  external  influence  which  can  be  brought  to 
bear  upon  them.  They  seem  to  be  entirely  determined  by  con- 
ditions within  the  core  of  the  atom.  Within  recent  years 
attempts  to  bring  about  a decomposition  of  certain  atoms,  by 
subjecting  them  to  powerful  electric  discharges  or  to  a bombard- 
ment by  the  rays  given  out  by  radium,  seem  to  have  met  with 
some  measure  of  success  and  it  may  eventually  prove  possible  to 
break  up  artificially  some  of  the  elements  into  the  simpler  constit- 
uents of  which  their  atoms  are  composed.  There  is  certainly 
nothing  in  our  present  knowledge  of  the  elements  to  justify  the 
view  that  such  a decomposition  is  impossible  although  it  will 
undoubtedly  be  very  difficult  and  will  require  the  use  of  agencies 
which  twenty  years  ago  were  entirely  unknown  to  science. 

(g)  Structure  of  the  Atom. — In  addition  to  the  violent  expul- 
sion of  electrons  which  is  characteristic  of  the  /5-ray  disintegration 
in  the  case  of  radioactive  elements,  the  atoms  of  all  elements  are 
able  to  lose  temporarily  or  under  certain  special  conditions  a 
limited  number  of  electrons  without  breaking  up  or  becoming  in 
any  way  unstable.  In  describing  some  of  the  various  circum- 
stances under  which  this  has  been  observed  to  occur  we  will  at 
the  same  time  interpret  the  known  facts  in  terms  of  an  hypothesis 
suggested  by  Sir  J.  J.  Thomson.®  Regardless  of  whether  this 
hypothesis  is  true  or  not  it  has  the  advantage  possessed  by  all 
good  hypotheses  of  serving  to  correlate  and  systematize  the  data 

“ Joseph  John  Thomson,  Kt.,  O.  M.,  F.  R.  S.  (1856-  ),  Cavendish 

Professor  of  Experimental  Physics  at  the  University  of  Cambridge  and 
Professor  of  Natural  Philosophy  at  the  Royal  Institution,  London.  He  is 
performing  wonders  in  revealing  the  inner  nature  of  the  atom. 


8 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


of  experiment  and  to  suggest  new  directions  of  investigation. 
According  to  the  views10  of  Sir  J.  J.  Thomson  the  atom  is  composed 
of  a core  of  positive  electricity  containing  a large  number  of 
electrons  which  are  very  firmly  held.  In  addition  to  these  central 
electrons  there  are  a smaller  number  near  the  surface  or  outer 
shell  of  the  atom  which  are  very  mobile  and  are  not  so  firmly  held. 
These  are  called  valence  electrons  and  their  number  determines 
the  atom’s  maximum  valency. 

When  one  atom  unites  with  another  to  form  a molecule,  the 
two  atoms  are  supposed  to  be  held  together  by  an  electrical 
attraction  due  either  to  the  passage  of  one  or  more  valence 
electrons  from  one  atom  to  the  other,  thereby  producing  a polar- 
ized molecule  with  one  atom  charged  positively  and  the  other 
negatively;  or  due  to  the  movements  of  valence  electrons  to  or 
about  certain  positions  in  their  own  atoms,  such  that  there  is 
a resulting  electrical  attraction  between  the  two  atoms  although 
they  both  remain  electrically  neutral  since  neither  has  gained  or 
lost  any  electricity.  The  molecule  of  hydrochloric  acid  is,  accord- 
ing to  Sir  J.  J.  Thomson’s  hypothesis,  an  example  of  a polarized 
molecule  formed  by  the  first  method.  Whether  this  hypothesis 
of  the  formation  of  the  hydrochloric  acid  molecule  be  correct  or 
not,  we  know  that  when  the  hydrochloric  acid  is  dissolved  in 
water  its  molecule  splits  up  in  such  a way  that  one  of  the  valence 
electrons  of  the  hydrogen  atom  remains  attached  to  the  chlorine 
atom,  which  is  therefore  negatively  charged,  while  the  hydrogen 
atom  which  has  lost  the  electron  becomes  thereby  positively 
charged.  Any  free  atom  or  molecule  which  carries  a charge  of 
electricity  is  called  an  ion  and  the  process  of  the  production  of 
ions  is  known  as  ionization.  In  the  case  cited,  the  ionization 
of  the  hydrochloric  acid  when  dissolved  in  water  is  spontaneous 
and  is  known  as  electrolytic  ionization  or  electrolytic  dissociation. 
Any  substance  which  ionizes  in  this  manner,  that  is,  by  solution 
in  a suitable  solvent,  is  called  an  electrolyte.  Another  type  of 
ionization  occurs  when  a gas  is  subjected  to  the  action  of  the  radi- 
ations given  off  by  radium  or  to  the  action  of  cathode  rays  or 
various  other  similar  agencies.  The  rapidly  moving  a-  and  p- 
particles  when  they  collide  with  the  atoms  of  the  gas  cause  them 
to  lose  temporarily  one  or  more  of  their  electrons  and  thus  to 
become  ions.  Again,  when  ultraviolet  light  is  allowed  to  fall 


Sec.  3] 


THE  STRUCTURE  OF  MATTER 


9 


upon  a metal  it  causes  the  metal  to  emit  electrons  and  thus 
to  acquire  a positive  charge.  Furthermore,  the  electrical  be- 
havior of  metals  indicates  that  some  of  the  electrons  are  able  to 
move  about  from  atom  to  atom  within  the  body  of  the  metal  with 
comparative  ease  so  that  when  an  electromotive  force  is  applied 
to  the  ends  of  a piece  of  metal  a stream  of  these  electrons  through 
the  metal  is  immediately  set  up.  These  moving  electrons  con- 
stitute the  electric  current  in  the  metal. 

(h)  Molecular  Motion. — The  molecules  of  every  substance, 
the  atoms  within  the  molecules  and  the  electrons  within  the  atoms 
are  in  constant  motion.  The  heat  content  of  any  body  consists 
of  the  kinetic  and  potential  energy  possessed  by  its  moving 
molecules  and  atoms.  The  motion  of  the  electrons  gives  rise 
to  radiant  energy,  including  light,  radiant  heat  and  Rontgen 
rays. 

3.  Hypothesis,  Theory,  Law  and  Principle.7 — In  the  preceding 
section  the  value  and  purpose  of  hypothesis  was  illustrated  in 
connection  with  the  discussion  of  the  structure  of  the  atom.  The 
terms  hypothesis  and  theory  are  frequently  used  more  or  less 
synonomously  but  the  latter  term  is  more  properly  employed  to 
designate  a system  which  includes  perhaps  several  related  hy- 
potheses together  with  all  the  logical  consequences  to  which  they 
lead,  the  whole  • serving  to  correlate  and  interpret  the  data  of 
experiment  in  some  particular  field  of  knowledge.  Thus  the 
hypothesis  that  gases  are  composed  of  a large  number  of  very 
small  particles,  together  with  certain  auxiliary  hypotheses 
regarding  the  shape,  motion  and  energy  of  these  particles  and  the 
forces  acting  between  them,  leads  to  a logical  system  by  means  of 
which  we  can  interpret  successfully  (■ i.e .,  “explain”)  the  known 
facts  concerning  the  behavior  of  gases.  This  system  is  called  the 
kinetic  theory  of  gases.  As  to  the  basic  hypothesis  that  gases 
are  composed  of  these  individual  particles  or  molecules,  it  has 
already  been  stated  (I,  1)  that  the  evidence  supporting  it  has 
recently  become  so  convincing  that  the  scientific  world  no  longer 
entertains  a reasonable  doubt  of  the  correctness  of  the  hypothe- 
sis. When  such  a condition  is  reached  the  hypothesis  is  consid- 
ered as  definitely  established  and  is  called  a fact.  There  is  evi- 
dently no  sharp  line  of  distinction  between  hypothesis  and  fact. 
It  is  simply  a question  of  degree  of  probability.  When  the  degree 


10 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


of  probability  of  the  correctness  of  a hypothesis  becomes  suffi- 
ciently high  it  may  be  regarded  as  a fact. 

The  term  law  of  nature  is  applied  to  a relation,  the  evidence  in 
support  of  which  is  so  strong  as  to  compel  a general  belief  in  its 
validity.  If  the  relation  is  a very  general  one  supported  by  a large 
and  varied  experience  so  that  the  chance  of  its  ever  being  found 
invalid  is  extremely  small  it  may  be  called  a principle. 

4.  The  Principle  of  the  Conservation  of  Matter. — This  princi- 
ple states  that  matter  can  neither  be  created  nor  destroyed. 
The  total  amount  of  matter  in  the  universe  remains  constant. 
The  most  exact  experimental  test  of  this  principle  for  a particular 
process  was  made  in  1908,  by  Landolt,®  who  showed,  in  the  case 
of  15  different  chemical  reactions,  that  the  quantity  of  matter 
(as  measured  by  its  weight)  before  and  after  the  occurrence  of 
the  reaction  was  in  every  case  constant  to  within  one  part  in  a 
million,  which  was  the  limit  of  accuracy  of  the  experiments. 
The  whole  experience  of  scientists  since  the  birth  of  experi- 
mental science  has  never  yielded  a single  instance  of  an  ex- 
ception to  this  principle  and  its  validity  cannot  be  reasonably 
doubted. 

5.  The  Law  of  Combining  Weights. — If  nA  atoms,  each  of 
mass,  raA,  of  the  element  A are  united  with  nB  atoms  each  of 
mass,  mB,  of  the  element  B to  form  the  molecule  of  a compound 
and  if  in  any  given  quantity,  M,  of  this  compound  there  are  nr 
molecules,  then  the  total  mass  of  the  element  A in  the  M grams  of 
the  compound  will  evidently  be  nAXmAXn',  that  of  the  ele- 
ment B will  be  nBXuiBXn'  and  the  ratio  of  the  two  masses  will 

, nAXmAXn'  nAXmA  , . . ' . 

be  — — 7 or  — bince  by  chemical  analysis  or 

nBXmBXn'  nBXmB  J 

synthesis  the  ratio  of  the  masses  of  any  two  elements  in  a chemical 

compound  can  be  very  accurately  determined,  we  can  in  this  way 

ascertain  very  exactly  the  numerical  value  of  the  ratio,  - > 

uB  x rnB 

which  is  called  the  combining  weight  ratio  for  the  two  elements, 
A and  B,  in  this  compound.  If  we  arbitrarily  agree  upon  some 
number  to  represent  the  combining  weight,  nAXmA,  of  the  ele- 

° Hans  Landolt  (1831-1910),  Professor  of  Chemistry  in  the  University 
of  Berlin  and  founder  of  the  Landolt-Bornstein,  Physikalisch-Chemische 
Tabellen. 


Sec.  5] 


THE  STRUCTURE  OF  MATTER 


11 


ment  A in  the  above  compound,  then  a value  for  the  combining 
weight,  nBX^B>  of  the  element  B can  evidently  be  readily 
calculated.  Then  by  determining  the  combining  weight  ratio 
for  a compound  of  a third  element  C with  either  A or  B a value 
for  the  combining  weight  of  this  third  element  can  also  be  calcu- 
lated. Proceeding  in  this  way  we  can  build  up  a table  of  com- 
bining weights  of  all  the  elements  capable  of  forming  compounds 
and  since,  with  the  exception  of  the  complicated  molecules  of 
certain  organic  compounds,  the  number  of  atoms  (nA,  nB,  etc.) 
in  any  molecule  is  comparatively  small,  it  is  evident  that  the  com- 
bining weights  thus  obtained  will  express  accurately  the  relative 
masses  of  the  elements  which  enter  into  chemical  combination 
with  one  another  in  all  chemical  reactions.  In  a similar  way  we 
can  show  that  the  masses  of  any  two  pure  chemical  substances 
whether  elements  or  compounds,  which  take  part  together  in 
any  chemical  reaction  must  stand  to  each  other  in  the  ratio  of 
small  whole  numbers,  this  ratio  being  their  combining  weight 
ratio  for  the  chemical  reaction  in  question. 

If  we  arbitrarily  adopt  the  number  8 as  the  combining  weight  of 
oxygen,  then  the  combining  weight  of  any  other  substance  may  be 
defined  as  that  weight  (in  grams)  of  it  which  combines  with  8 
grams  of  oxygen;  or,  if  the  substance  does  not  combine  with 
oxygen,  then  that  weight  of  it  which  combines  or  reacts  with  one 
combining  weight  of  any  other  substance  will  be  its  combining 
weight.  With  this  definition  of  the  term,  combining  weight,  we 
may  state  the  Law  of  Combining  Weights  in  the  following  terms: 
Pure  chemical  substances  react  with  one  another  only  in  the 
proportions  of  their  combining  weights.  If  an  element  forms 
several  compounds  with  oxygen  it  may  have  several  combining 
weights.  Thus  in  the  following  compounds,  N20,  NO,  N203, 
N02,  and  N205,  the  number  of  grams  of  nitrogen  combined 
with  8 grams  of  oxygen  is  14, 7, 4f , 3 J,  and  2f  grams  respectively. 
Of  these  numbers  we  may,  if  we  wish,  choose  any  one  and  call  it 
the  combining  weight  of  nitrogen.  Thus  if  we  choose  the  num- 
ber 7,  then  the  other  numbers  are  respectively  2,  f , \ and  f times 
the  combining  weight  of  nitrogen,  as  is  required  by  the  law  of 
combining  weights. 

This  law  which  we  have  here  shown  to  be  a necessary  corollary 
of  the  atomic  and  molecular  structure  of  matter  was  discovered 


12 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


by  Richter0  before  the  atomic  theory  was  proposed  by  Dalton  in 
1808.  In  fact  in  the  case  of  most  of  the  laws  and  principles  which 
in  the  following  pages  we  shall  show  to  be  direct  consequences  of 
the  atomic  and  molecular  structure  of  matter,  the  law  or  principle 
in  question  was  discovered  empirically  before  the  structure  of 
matter  was  known  and  in  some  instances  even  before  the  modern 
atomic  and  molecular  theories  had  been  proposed.  One  of  the 
triumphs  of  these  theories  was  their  ability  to  interpret  all  of 
these  empirical  laws  from  a single  point  of  view. 

Problem  1. — The  following  problem  will  illustrate  the  degree  of  accuracy 
with  which  the  law  of  combining  weights  has  been  established:  The  follow- 
ing two  sets  of  combining  weight  ratios  have  been  determined  by  chemical 
analysis  and  synthesis: 

(a) 

=0.849917 
0.646230 

I2W5 

Calculate  from  each  set  of  data  a value  for  the  combining  weight  of  silver, 
taking  16  as  the  combining  weight  of  oxygen. 


Li  CIO  4 
LiCl 
AgCl 
LiCl 
Ag 
LiCl 


(b) 

= 2.5097 
= 3.3809 
= 2.5446 


6.  Atomic  Weights. — It  is  evident  from  the  preceding  that  if 
the  weight  of  a single  atom  of  any  element  is  known,  then  the 
weight  of  an  atom  of  any  other  element  which  combines  with 
this  one  can  be  calculated,  provided  the  number  of  atoms  of  each 
element  in  the  molecule  of  the  compound  is  known  and  provided 
the  combining  weight  ratio  of  the  two  elements  in  this  compound 
has  been  determined.  For  example,  the  weight  of  an  atom  of  oxy- 
gen is  known  to  be  26.39  X 10-24  grams.  The  molecule  of  water 
is  known  to  consist  of  two  atoms  of  hydrogen  combined  with  one 

of  oxygen  and  the  combining  weight  ratio,  has  been 

found  by  chemical  synthesis  to  be  0.12594.  We  have,  therefore, 
2 X 'win 

1 ^<26.39 xlO^  = 1^594:  and  hence  raH,  the  weight  of  an  atom 

of  hydrogen,  must  be  16.6  X 10~25  grams.  Proceeding  in  this  way 
we  could  compute  a table  of  the  weights  of  the  atoms  of  all 


0 Jeremias  Benjamin  Richter  (1762-1807).  Chemist  at  the  Berlin 
Porcelain  Factory  and  Assessor  for  the  Prussian  Bureau  of  Mines.  He 
determined  the  first  set  of  equivalent  weights  for  the  metals. 


Sec.  6] 


THE  STRUCTURE  OF  MATTER 


13 


elements  capable  of  forming  compounds.  The  practical  objec- 
tion to  such  a table  is  that  while  the  relative  weights  of  the  atoms 
can  be  very  accurately  determined  (to  0.01  per  cent,  in  many 
cases)  by  chemical  analysis  and  synthesis,  the  actual  weight  of 
any  atom  has  not  yet  been  determined  to  better  than  0.2  per 
cent,  and  consequently  all  the  values  of  such  a table  would  be 
subject  to  frequent  revision  as  our  knowledge  of  the  weight  of 
the  atom  of  some  element  became  more  exact.  This  objection 
can  be  avoided  by  arbitrarily  adopting  any  desired  number  as  the 
atomic  weight  of  some  one  element  and  then  computing  the 
relative  atomic  weights  of  the  others  from  their  experimentally 
determined  combining  weights  and  a knowledge  of  the  formula 
of  the  compound  in  question.  Chemists  have  agreed  to  adopt 
the  number  16  as  the  atomic  weight  of  the  element,  oxygen,  and 
as  the  basis  of  the  atomic  weight  table.1  In  order  to  compute 
the  actual  weight  of  any  atom  from  the  atomic  weight  of  the 
element  it  is  only  necessary  to  divide  the  latter  number  by 

26.39  X10~24  = 60'62  X 1Q22’ 

the  number  of  atoms  in  one  atomic  weight  of  any  element.  This 
quantity  is  one  of  the  universal  constants  of  nature  and  is  known 
as  Avogadro’s  number.  We  shall  represent  it  by  the  symbol,  N. 

We  have  seen  that  if  an  element  forms  several  compounds  with 
oxygen  it  may  have  several  combining  weights.  One  of  these 
combining  weights  or  some  submultiple  of  one  of  them  will  also 
be  the  atomic  weight.  It  must  evidently  be  that  one  which 

1 The  modern  atomic  weight  table  was  formerly  based  upon  the  value 
unity  which  was  arbitrarily  taken  as  the  atomic  weight  of  hydrogen.  On 
this  basis  the  atomic  weight  of  oxygen  was  found  to  be  slightly  less  than  16. 
The  atomic  weights  of  many  of  the  other  elements  were  based  upon  combin- 
ing weight  ratios  which  involved  either  directly  or  indirectly  a knowledge 
of  the  atomic  weight  of  oxygen  and  this  in  turn  was  based  upon  the  experi- 
mentally determined  value  of  the  combining  weight  ratio  of  oxygen  to  hydro- 
gen. The  result  was  that  whenever  new  determinations  resulted  in  a 
change  or  a more  accurate  knowledge  of  this  latter  ratio,  it  became  necessary 
to  recalculate  a large  part  of  the  atomic  weight  table.  In  order  to  avoid 
this  it  was  decided  to  arbitrarily  adopt  16  as  the  atomic  weight  of  oxygen 
and  to  employ  the  combining  weight  ratio  of  oxygen  to  hydrogen  simply 
to  determine  the  atomic  weight  of  hydrogen.  The  atomic  weights  of  the 
other  elements  are  thus  not  affected  by  changes  in  the  value  of  this  ratio. 


14 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


expressed  in  grams  contains  the  same  number  of  atoms  as  16 
grams  of  oxygen.  Methods  for  deciding  which  of  several  com- 
bining weights  is  the  atomic  weight  of  an  element  will  be  discussed 
later  (II,  12). 

The  International  Committee  on  Atomic  Weights  computes 
each  year  a table  of  atomic  weights  based  upon  the  most  reliable 
determinations.  This  table  is  published  each  year  in  all  of  the 
leading  chemical  journals  of  the  world.  The  International 
Atomic  Weight  Table  for  1915  is  given  on  the  opposite  page. 

7.  Chemical  Formulas,  Formula  Weights,  Equivalent  Weights 
and  Molecular  Weights. — The  formula  of  any  pure  chemical 
substance  is  a succession  of  the  symbols  of  the  elements  it  con- 
tains, each  symbol  being  provided  with  such  integers  as  subscripts 
as  will  make  the  resulting  formula  express  the  atomic  proportions 
of  the  elements  in  the  compound.  In  addition  the  symbol  of 
each  element  represents  one  atomic  weight  in  grams  of  that 
element  and  the  whole  formula  represents  a weight  in  grams  of 
the  substance  which  is  equal  to  the  sum  of  all  the  atomic  weights 
each  multiplied  by  its  subscript.  This  weight  is  called  the  gram- 
formula  weight  of  the  substance.  Thus  the  formula,  HN03, 
denotes  1.008+14.01  + 3X16  = 63.02  grams  of  nitric  acid  and 
the  formula,  JAs203,  represents  | (2  X 74.96 +3  X 16)  = 98.96 
grams  of  arsenic  trioxide.  That  weight  in  grams  of  any  substance 
which  reacts  chemically  with  one  gram-atomic  weight  of 
hydrogen,  or  with  that  amount  of  any  other  substance  which 
itself  reacts  with  one  gram-atomic  weight  of  hydrogen,  is  called 
the  equivalent  weight  or  one  equivalent  of  the  substance.  Thus 
the  equivalent  weight  of  each  of  the  following  substances  is 
the  gram-formula  weight  indicated:  |C12,  |02,  Ag,  JZn,  JBi, 
JBa(OH)2,  |H2S04,  iH3P04,  -|A1C13,  }K4Fe(CN)6.  The  same 
substance  may  have  more  than  one  equivalent  weight  depend- 
ing on  whether  the  reaction  with  hydrogen  is  one  of  metathesis 
or  of  oxidation  and  reduction.  Thus  the  metathetical  equiva- 
lent of  ferric  chloride  is  | FeCl3,  but  its  oxidation  equivalent 
(when  reduced  ferrous  chloride)  is  FeCl3  and  the  metathetical 
equivalent  of  potassium  chlorate  is  KC103  while  its  oxidation 
equivalent  (when  reduced  to  KC1)  is  JKC103. 

The  molecular  formula  of  a substance  is  the  formula  which 
expresses  the  atomic  composition  of  the  molecule  and  the  molecu- 


Sec.  7] 


THE  STRUCTURE  OF  MATTER 


15 


Table  I. — International  Atomic  Weights,  1915 


Symbol 

' 

Atomic 

weight 

Symbol 

Atomic 

weight 

A1 

27.1 

Molybdenum 

Mo 

96.0 

Sb 

120.2 

Neodymium 

Nd 

144.3 

A 

39.88 

Neon 

Ne.  . . 

20.2 

As 

74.96 

Nickel 

Ni 

58.68 

Barium 

Ba 

137.37 

Niton  (radium  emanation) . . . ! 

Nt 

222.4 

Bismuth 

Bi 

208.0 

Nitrogen 

N 

14.01 

B 

11.0 

Osmium 

Os 

190.9 

Br..  .. 

79  92 

Oxygen  . . 

0 

16.00 

Cd 

112.40 

Palladium 

Pd.  . . 

106.7 

Cs 

132.81 

Phosphorus 

P 

31.04 

Ca 

40.07 

I Platinum 

Pt 

195.2 

c 

12.00 

Potassium 

K 

39.10 

Ce 

140.25 

Praseodymium 

Pr 

! 140.6 

Cl 

35.46 

Radium 

Ra 

226.0 

Cr 

52.0 

Rhodium 

Rh 

102.9 

Cobalt 

Co 

58.97 

Rubidium 

Rb 

85.45 

Columbium 

Cb 

93.5 

' Ruthenium 

Ru 

101.7 

Copper 

Cu 

63.57 

Samarium 

Sa 

150.4 

Dysprosium 

Dy 

162.5 

Scandium 

Sc. ... .. 

44.1 

Erbium 

Er 

167.7 

Selenium 

Se 

79.2 

Europium 

Eu 

152.0 

Silicon 

Si 

28.3 

Fluorine 

F 

19.0 

Silver 

Ag 

107.88 

Gadolinium 

Gd 

157.3 

Sodium 

Na 

23.00 

Gallium 

Ga 

69.9 

Strontium 

Sr 

87.63 

Germanium 

Ge 

72.5 

Sulphur 

S 

32.07 

Glucinum . . . 

Gl.. . . 

9.1  ' 

Tantalum . 

Ta. . . 

181.5 

Gold 

Au ! 

197.2 

T ellurium 

Te 

127.5 

Helium 

He 

3.99 

Terbium 

Tb 

159.2 

Holmium 

Ho 

163.5 

Thallium 

T1 

204.0 

Hydrogen 

H 

1.008 

Thorium 

Th 

232.4 

Indium 

In 

114.8 

Thulium 

Tm 

168.5 

Iodine 

I. . . 

126.92 

Tin 

Sn 

119.0 

Iridium 

Ir 

193.1 

Titanium 

Ti 

48.1 

Iron 

Fe j 

55 . 84 

Tungsten 

w ! 

184.0 

Krypton 

Kr 

82.92 

Uranium 

u 

238.5 

Lanthanum 

La 

139.0 

Vanadium 

V 

51.0 

Lead . . 

Pb 

207. 10 

Xenon. . 

Xe  . 

130.2 

Lithium . . 

Li 

6.94 

Ytterbium  (Neoytterbium) 
Yttrium 

Yb  . 

173.0 

Lutecium 

Lu 

174.0 

Yt 

89.0 

Magnesium 

Mg 

24.32 

Zinc 

Zn 

65.37 

Manganese 

Mn 

54.93 

Zirconium 

Zr 

90.6 

Mercury 

Hg 

200.6 

lar  weight  (more  properly,  the  gram-molecular  weight  or  the 
molal  weight)  is  the  weight  in  grams  indicated  by  the  molecular 
formula.  Thus  the  molecular  formula  of  gaseous  hydrogen, 
whose  molecules  are  diatomic,  is  H2  and  its  gram -molecular  weight 
is  2X  1.008  = 2.016  grams.  The  molecular  formulas  of  some  of 
the  other  elements  in  the  gaseous  state  are  as  follows:  N2,  Fe, 
Cl2,  Br2,  I2,  P4,  AS4,  He,  A,  Hg,  Cd,  Zn.  The  molecular  weight 


16 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  I 


evidently  bears  the  same  relation  to  the  weight  of  the  molecule 
that  the  atomic  weight  does  to  the  weight  of  the  atom,  that  is, 
the  former  is  in  each  instance  N-times  the  latter.  One  gram 
molecular  weight  of  a substance  is  frequently  called  one  mole. 

8.  States  of  Aggregation  and  Phases. — Matter  occurs  in  differ- 
ent states  or  conditions  known  as  states  of  aggregation.  The 
three  principal  states  of  aggregation  are  the  gaseous  state,  the 
liquid  state  and  the  crystalline  state.  A body  is  said  to . be 
isotropic  when  it  displays  the  same  physical  properties  in  all 
directions  through  it.  All  gases,  most  liquids,  and  the  so-called 
amorphous  solids  are  isotropic.  Anisotropic  bodies  display 
different  physical  properties  in  different  directions.  They  com- 
prise nearly  all  crystalline  substances.  Another  distinction 
between  these  two  classes  appears  when  we  consider  the  passage 
of  a substance  from  one  state  of  aggregation  to  another.  The 
passage  from  one  isotropic  state  to  another  may  be  either  con- 
tinuous or  discontinuous  while  the  passage  from  the  anisotropic 
to  the  isotropic  state,  or  vice  versa , has  thus  far  been  found  to  be 
always  a discontinuous  process. 

Any  portion  of  the  universe  which  we  choose  to  separate  in 
thought  from  the  rest  of  the  universe  for  the  purpose  of  consider- 
ing and  discussing  the  various  changes  which  may  occur  within 
it  under  various  conditions  is  called  a system  and  the  rest  of  the 
universe  becomes  for  the  time  being  the  surroundings  of  the 
system.  Thus,  if  we  wish  to  consider  the  changes  which  occur 
when  salt  and  water  are  brought  together  under  various  condi- 
tions of  temperature,  pressure,  etc.,  then  the  two  spbstances 
salt  and  water  constitute  our  system  and  they  are  called  the 
components  of  the  system.  The  physically  homogenous  but 
mechanically  separable  portions  of  a system  are  called  its  phases. 
Thus  a system  containing  crystals  of  benzene,  liquid  benzene, 
and  benzene  vapor  contains  three  phases,  one  crystalline  phase, 
one  liquid  phase  and  one  gaseous  phase.  If  we  add  salt  to  this 
system  we  may  have  four  phases,  i.e .,  two  crystalline  phases 
(usually  called  solid  phases),  namely,  the  salt  crystals  and  the 
benzene  crystals,  one  liquid  phase  composed  of  a solution  of  salt 
in  benzene  and  one  gaseous  phase  composed  chiefly  of  benzene 
vapor.  Each  phase  is  distinguishable  from  the  others  by  being 
separated  from  them  by  definite  and  sharp  bounding  surfaces 


Sec.  9] 


THE  STRUCTURE  OF  MATTER 


17 


and  by  being  itself  homogeneous  throughout  its  own  interior. 
By  homogeneous  is  meant  of  uniform  chemical  composition 
throughout  and  having  the  same  physical  properties  in  all  re- 
gions within  itself.  A system  composed  of  only  one  phase  is, 
therefore,  a homogeneous  system,  while  one  composed  of  more 
than  one  phase  is  called  a heterogeneous  system.  The  dis- 
tinction between  phase  and  state  of  aggregation  should  be  clearly 
understood.  A system  composed  of  liquid  water  and  liquid 
mercury  has  two  phases  but  only  one  state  of  aggregation,  the 
liquid  state.  Since  all  gases  are  miscible  with  one  another  in 
all  proportions  there  can  never  be  more  than  one  gaseous  phase 
in  any  system.  There  may,  however,  be  several  crystalline 
and  several  liquid  phases  present,  if  the  system  contains  a suffi- 
cient number  of  components.  The  condition  of  any  phase  of  a 
system  which  has  reached  a state  of  equilibrium  is  ordinarily 
completely  determined  if  its  pressure,  p,  temperature,  T,  and 
composition,  C,  are  fixed,  or  in  mathematical  language,  if 

Up,  T,  o-o  (1) 

The  particular  function  which  connects  the  pressure,  tempera- 
ture and  composition  of  any  phase  is  called  the  equation  of  state 
or  the  condition  equation  of  that  phase. 

For  every  heterogeneous  system  which  has  reached  a state  of 
equilibrium  under  a given  set  of  conditions,  there  exists  a rela- 
tion, known  as  the  Phase  Rule,  which  connects  the  number  (p) 
of  phases  present  in  the  system,  with  the  number  (c)  of  its  com 
ponents  and  with  the  number  (f)  of  variables  (temperature, 
pressure  and  percentage  composition  of  the  phases)  whose  values 
must  be  known  before  the  condition  of  the  system  becomes  com- 
pletely determined.  This  relation  is  expressed  by  the  equation 

P + f = c + 2 (2) 

By  c,  the  number  of  components,  is  meant  the  smallest  number 
of  substances  by  which  the  percentage  composition  of  each  phase 
in  the  system  can  be  expressed.  The  derivation  of  the  Phase 
Rule  and  a more  precise  definition  of  the  quantities  involved 
in  it  will  be  given  in  a later  chapter. 

9.  Chemical  Equilibrium. — When  a system  contains  molecular 
species  which  are  reacting  chemically  with  one  another  to  produce 


18 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


other  molecular  species  which  in  turn  are  similarly  constantly 
reacting  with  one  another  to  reproduce  the  first  ones  again,  so 
that  the  rates  of  reaction  in  the  two  directions  are  equal  and  hence 
the  relative  amounts  of  the  different  molecular  species  concerned 
remain  constant,  a chemical  equilibrium  is  said  to  exist  within 
the  system.  The  general  method  for  representing  a chemical 
equilibrium  will  be  by  means  of  the  following  equation: 

aA+frB-f  . . . <=±  mM+^N+  ....  (3) 

which  represents  a reaction  in  which  a molecules  of  the  substance 
A,  react  with  b molecules  of  the  substance,  B,  etc.,  to  form  m 
molecules  of  the  substance,  M,  and  n molecules  of  the  substance, 
N,  etc.,  and  vice  versa. 


REFERENCES 

Books:  (2)  The  Study  of  the  Atom.  F.  P.  Venable,  1904.  (3)  Radio- 
active Substances  and  Their  Radiations.  E.  Rutherford,  1913.  (4)  Rays 

of  Positive  Electricity  and  Their  Application  to  Chemical  Analysis.  Sir  J.  J. 
Thomson,  1914.  (5)  The  Chemistry  of  the  Radio  Elements.  F.  Soddy, 
Part  I,  1911.  (6)  Ibid.,  Part  II,  1914.  (7)  Science  and  Hypothesis. 
H.  Poincare,  1907. 

Journal  Articles:  (8)  Richards  and  Lembert,  Jour.  Amer.  Chem.  Soc., 
36,  1329  (1914).  (9)  Tolman,  Osgerby  and  Stewart,  Ibid,  36,  485  (1914). 

(10)  Thomson,  Phil.  Mag.  27,  757  (1914).  (11)  Eve,  Modern  Views  on 

the  Constitution  of  the  Atom.  A Review.  Science,  40,  115  (1914). 


CHAPTER  II 


THE  GASEOUS  STATE  OF  AGGREGATION1 


1.  Definition  and  Structure  of  a Gas. — A substance  is  in  the 
gaseous  state,  if  it  remains  homogeneous  and  its  volume  increases 
without  limit  when  the  pressure  upon  it  is  continuously  decreased 
and  its  temperature  is  kept  constant.  In  gases  under  low  pres- 
sures the  molecules  are  so  far  apart  and  occupy  such  a small  frac- 
tion of  the  total  volume  of  the  containing  vessel  that  they  are 
nearly  independent  of  one  another.  They  are  in  constant  and 
very  rapid  motion  in  all  directions,  moving  in  nearly  straight 
lines  and  colliding  frequently  with  one  another  and  with  the  walls 
of  the  containing  vessel.  They,  therefore,  describe  zig-zag  paths 
of  such  complicated  and  uncertain  characters  owing  to  the  nu- 
merous collisions  which  they  experience,  that  it  is  not  possible  to 
predict  where  a given  molecule  will  be  after  an  interval  of  time. 
Motion  of  this  character  is  called  unordered  or  random  motion. 
The  collisions  between  the  molecules  of  a given  gas  will  evidently 
be  more  frequent  the  greater  the  number  of  molecules  in  a given 
volume  of  the  gas,  that  is,  the  greater  the  density  of  the  gas.  The 
average  distance  through  which  a molecule  travels  between  two 
successive  collisions  is  called  the  mean  free  path  of  the  molecule. 
The  collisions  between  molecules  and  between  a molecule  and  the 
wall  of  the  containing  vessel  take  place  without  loss  of  energy, 
that  is.  they  are  perfectly  elastic  collisions.  The  mean  kinetic 
energy,  \ mu2,  of  the  molecules  of  a perfect  gas  has  been  shown 
experimentally  to  depend  only  upon  the  temperature  and  to  be 
independent  of  the  nature  of  the  gas.  (IX,  4) 

2.  Boyle’s  Law. — The  pressure  ( i.e .,  the  force  per  unit  surface) 
exerted  by  a gas  upon  the  walls  of  the  containing  vessel  is  due 
to  the  impacts  of  the  rapidly  moving  molecules  and  since  by 
definition 


/ 


— ma  — m 


d u 
df 


_ d^ 
— dt 


{mu) 


19 


(1) 


20 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


where  t signifies  time,  this  pressure  will  be  determined  by  the 
time  rate  of  change  of  momentum,  mu,  which  the  molecules 
experience  on  striking  the  walls. 

In  order  to  calculate  this  pressure  we  may  assume  that  the 
molecules  (which  in  reality  are  constantly  changing  their  actual 
velocities  after  each  impact)  are  all  moving  with  the  same  veloc- 
ity, u,  whose  square  is  equal  to  the  mean  of  the  squares  of  the 
actual  velocities.  In  other  words,  the  velocity,  u,  has  such  a 
value  that  if  all  the  molecules  possessed  it,  the  mean  molecular 
kinetic  energy  \mu 2,  would  be  unchanged.  It  is  evident  that 
the  pressure  would  also  be  unchanged.  This  assumption  is 
made  for  the  purpose  of  simplifying  our  calculation  which  may, 
however,  be  carried  through  to  the  same  conclusion  without 
making  this  simplifying  assumption.  Since  the  pressure  exerted 
by  the  gas  is  evidently  independent  of  the  shape  of  the  vessel 
containing  it,  we  shall  assume  for  convenience  that  it  is  con- 
tained in  a cube  of  side,  l,  and  we  shall  consider  the  three  com- 
ponents of  the  velocity,  u,  which  are  respectively  perpendicular 
to  the  faces  of  the  cube  and  are,  therefore,  connected  with  the 
velocity,  u,  by  the  relation, 

U!2  + U22  + U32  = U2  (2) 

A molecule  of  mass,  m,  approaches  face,  1,  with  the  momen- 
tum. mui,  perpendicular  to  this  face  and  after  the  impact 
it  recedes  from  it  with  the  momentum,  —muh  the  change  in 
momentum  being,  therefore,  2mu\.  The  number  of  impacts  in 

unit  time  will  evidently  be  ^ and  the  total  change  of  momen- 
tum per  unit  time  will  be  the  product  of  the  number  of  impacts 

. . . . _ Ui  mu\2 

into  the  change  of  momentum  per  impact,  or  = i • 

Similarly  in  the  other  two  directions  the  total  change  in  momen- 

TYlUs^ 

turn  on  each  face  will  be  ^ and  j respectively,  and  the 

total  effect  of  this  molecule  upon  all  six  walls  of  the  cube  will 
therefore  be 


2 m 
l 


(ui2+u22+u32) 


2 mu2 
l 


(3) 


If  there  are  n molecules  of  gas  in  the  cube,  the  total  force  exerted 


Sec.  3] 


THE  GASEOUS  STATE  OF  AGGREGATION 


21 


by  the  gas  upon  all  the  walls  of  the  cube  will  be 


2 nmu2 

T 


and  the 


force  per  unit  area  or  the  pressure  will  be 

2 nmu2  nmu 2 nmu2  . . 

p = l -T-o t2  — 3^3  = 2>V  or  pv  = \nmul 


(4) 


where  v is  the  volume  in  which  the  molecules  are  free  to  move, 
here  assumed  equal  to  the  volume  of  the  cube. 

Since  at  constant  temperature  the  mean  kinetic  energy  and 
hence  mu2  is  a constant  (II,  1)  it  follows  from  equation  (4)  that 

pv  = const.  (5) 


for  a given  mass  of  any  gas  in  which  n does  not  change  with  p . 
This  is  Boyle’s  or  Mariotte’s  Law.  It  was  discovered  empirically 
by  Robert  Boyle0  in  1662  and  by  Mariotte  b in  1679. 

3.  Gay  Lussac’s  Law  of  Temperature  Effect. — Equation  (4) 
may  be  written 


pv  = \n  (%mu2)  (6) 

and  since  the  mean  kinetic  energies  of  the  molecules  of  all  gases 
are  identical  at  the  same  temperature,  it  follows  that  the  rate 
of  change  of  the  kinetic  energy  with  the  temperature  must  also 
be  the  same  for  all  gases,  for  otherwise  if  the  mean  kinetic  energies 
of  several  gases  were  all  equal  at  one  temperature,  they  could 
not  be  so  at  another.  In  mathematical  language  this  statement 
is  expressed  thus, 


d (\mu2) 

AQ 


= const,  (for  all  gases) 


(7) 


where  6 is  the  temperature, 
we  obtain 

d (pv)  2 fb(\mu2) 

AO  ~ *n  \~br 


Differentiating  equation  (6)  above 


l(\mu2) 


l dn 
\dd 


), 


mu * 


(8) 


° Robert  Boyle  (1627-1691).  Seventh  son  and  fourteenth  child  of 
Richard  Boyle,  first  Earl  of  Cork.  Educated  at  Eton  and  on  the  Continent. 
Settled  in  Oxford  where  he  erected  a laboratory.  One  of  the  founders  of  the 
Royal  Society  of  London.  A man  of  insatiable  curiosity  concerning  all 
kinds  of  natural  phenomena. 

b Edme  Mariotte  (1620-1684).  The  father  of  experimental  physics  in 
France.  His  treatise  on  The  Flow  of  Water  and  Other  Liquids  appeared  in 
1686.  His  collected  works  were  published  in  Leyden  in  1717  and  at  the 
Hague  in  1740.  He  was  one  of  the  earliest  members  of  the  French  Academy 
of  Sciences. 


1 


22 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY 


[Chap.  II 


Combining  this  with  equation  (7)  we  have 
d (pv) 


d-A  c)n 

^ = nX  const. + const.  X ^ 


(9) 


If  n is  a constant  with  respect  to  variations  in  9,  this  relation 
becomes 

) = const.  = k . oa,  j'  r (10) 

where  k is  a constant  independent  of  the  nature  of  the  gas  and 
dependent  only  on  the  initial  value  of  the  pressure  volume 
product.  This  result  may  be  stated  in  words  as  follows:  The 

temperature  rate  of  change  of  the  pressure -volume  product 
is  the  same  for  all  gases  whose  molecular  complexity  does 
not  change  with  the  temperature.  This  law  was  discovered 
empirically  by  Gay  Lussaca  in  1802.  It  may  be  expressed  more 
elegantly  and  concisely  by  the  differential  equation 


d2(p*>)  _ n 
d02  -u 


(11) 


of  which  equation  (10)  is  the  first  integral. 

4.  The  Law  of  Avogadro. — Equation  (6)  may  be  written 


_ 3 vv 

n 2 ( \mu 2) 


(12) 


from  which  it  follows  that  if  p,  v,  and  § mu2,  and  hence  also  9 
are  all  constants,  n is  likewise  a constant  and  has  the  same  value 
for  all  gases;  or  in  other  words  that  equal  volumes  of  all  gases 
at  the  same  temperature  and  pressure  contain  the  same  number 
of  molecules.  This  statement,  advanced  as  a hypothesis  in 
1811  by  the  Italian  physicist  Avogadro,6  is  of  great  importance 
in  the  determination  of  the  molecular  weights  of  gases.  The 


“ Louis  Joseph  Gay-Lussac  (1778-1850).  Studied  in  the  Ecole  Poly- 
technique at  Paris  under  Berthollet  and  Laplace  in  1797.  His  papers  on 
the  properties  of  gases  appeared  1801-1808.  He  was  the  discoverer  of 
cyanogen  and  the  inventor  of  many  accurate  methods  of  chemical  analysis. 

b Amedeo  Avogadro  (1776-1856).  Studied  law  and  became  a practising 
lawyer.  In  1800  began  the  study  of  mathematics  and  became  Professor 
of  Physics  at  Vercelli  and  later  at  Turin.  Avogadro’s  hypothesis  although 
published  in  1811  began  to  find  acceptance  among  chemists  only  after  1860, 
and  as  late  as  1885,  French  chemists  still  refused  to  accept  it  as  the  logical 
basis  for  determining  molecular  formulas. 


Sec.  5] 


THE  GASEOUS  STATE  OF  AGGREGATION 


23 


molecule  of  oxygen  is  known  to  have  the  formula  02  and  hence 
the  molecular  weight,  2X16  = 32.  Avogadro’s  number,  the 
number  of  molecules  in  32  grams  of  oxygen,  is  N = 60.62  X1022 
as  stated  in  the  first  chapter.  According  to  Avogadro’s  law 
this  is  also  evidently  the  number  of  molecules  in  one  gram- 
molecular  weight  of  any  gas  and  the  gram-molecular  weight  of  any 
gas  must,  therefore,  be,  according  to  Avogadro’s  law,  that  mass 
in  grams  of  the  gas  which  occupies  the  same  volume  as  do  32 
grams  of  oxygen  at  the  same  temperature  and  pressure. 

If  in  equation  10  of  the  preceding  section  we  agree  always  to 
take  one  gram -molecular  weight  of  a gas,  then  by  Avogadro’s  law 
we  have  for  every  gas 


where  ^0  is  the  volume  of  one  mole  of  the  gas  and  R 
is  a constant  whose  value  is  the  same  for  all  gases.  The 
product,  pv 0,  has  the  dimensions  of  force  X distance  and  hence 
of  work  or  energy.  (Problem:  Demonstrate  this.)  It  repre- 
sents the  work  required  to  produce  the  volume,  Vo,  against  the 
constant  pressure,  p,  and  may  be  called  the  molal  volume 
energy  of  the  gas  and  the  significance  of  equation  (13)  may  be 
expressed  in  the  following  words : The  temperature  rate  of  change 
of  the  molal  volume  energy  of  a pure  gas  is  independent  of 
the  temperature  and  of  the  nature  of  the  gas. 

5.  Definition  of  the  Centigrade  Degree  and  of  Absolute  Tem- 
perature.— The  centigrade  degree  is  arbitrarily  defined  as  T-^ 
part  of  the  temperature  interval  between  the  temperature  of 
ice,  melting  under  the  pressure  of  one  atmosphere,  and  the  tem- 
perature of  the  vapor  of  water,  boiling  under  the  pressure  of 
one  atmosphere.  The  size  of  this  degree  will  depend  somewhat 
upon  the  nature  of  the  material  composing  the  thermometer 
employed  in  measuring  this  temperature  interval.  Thus  on 
the  international  hydrogen  scale  the  centigrade  degree  is  defined 
as  that  difference  in  temperature  which  produces  in  the  pressure 
of  a quantity  of  hydrogen  gas  tto  part  of  the  change  in  pressure 
which  is  produced  when  the  volume  of  the  gas  is  kept  constant 
and  its  temperature  is  changed  from  that  of  ice  melting  under  a 
pressure  of  one  atmosphere  to  that  of  the  vapor  of  water  boiling 


24 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


under  a pressure  of  one  atmosphere,  it  being  further  stipulated 
that  the  hydrogen  shall  be  under  a pressure  of  1 meter  of  mercury 
when  it  is  at  the  temperature  of  melting  ice.  If  nitrogen  be 
employed  instead  of  hydrogen,  the  centigrade  degree  defined  in 
a similar  manner  is  almost  identical  with  that  of  the  hydrogen 
scale  and  as  the  initial  pressure  of  the  gas  employed  is  decreased, 
the  size  of  the  degree  as  defined  above  reaches  a limiting  value 
which  is  entirely  independent  of  the  nature  of  the  gas  employed 
in  the  thermometer.  This  degree  defines  a temperature  scale 
known  as  the  Kelvin  ° or  Thermodynamic  Scale  and  may  be  re- 
garded as  the  degree  given  by  a thermometer  filled  with  a perfect 
gas  (II,  7 and  9) . The  degree  on  the  Kelvin  scale  differs  from  that 
on  the  international  hydrogen  scale  by  less  than  0.1  per  cent, 
so  that  in  nearly  all  cases  the  two  scales  may  be  regarded  as  iden- 
tical for  practical  purposes.  Strictly,  however,  we  shall  under- 
stand the  centigrade  degree  to  be  that  defined  by  the  Kelvin 
scale. 

Equation  (10)  may  be  written,  d(py)  = kdd  and  on  integration 
gives 

pv  = kd-\~kk'  = k(0-\~k'^  (14) 

where  kk'  is  the  integration  constant.  In  words  this  equation 
states  that  the  pressure-volume  product  of  a given  mass  of  gas 
is  proportional  to  the  temperature-plus-a-constant,  k'.  The 
numerical  value  of  this  constant  depends  upon  the  size  of  the 
degree  on  our  temperature  scale.  If  we  adopt  the  centigrade 
degree  as  defined  above,  then  we  find  by  experiment  with  dif- 
ferent gases  that  the  constant  k'  has  the  value  273.1  centigrade 
degrees.  Equation  (14)  may,  therefore,  be  written 

pv  = k(t+ 273.1)  (15) 

and  the  symbol,  t , will  henceforth  be  understood  to  signify  tem- 
perature on  the  centigrade  scale. 

a William  Thomson,  Lord  Kelvin  (1824-1907).  Studied  at  the  Univer- 
sities of  Glasgow  (1834)  and  Cambridge  (1841).  Professor  of  Natural 
Philosophy  at  Glasgow,  1846.  Laid  the  first  Atlantic  cables  (1857). 
Knighted  in  1866  and  created  Baron  Kelvin  of  Largs,  1892.  Buried  in 
Westminster  Abbey. 


Sec.  5] 


THE  GASEOUS  STATE  OF  AGGREGATION 


25 


The  quantity,  £+273.1,  is  called  the  absolute  temperature 
and  is  represented  by  the  letter  T.  The  absolute  scale  evidently 
differs  from  the  centigrade  scale  only  in  having  its  zero  point 
273.1°  below  the  centigrade  zero.  Equation  (15)  may  now  be 
written 

pv  = kT  (16) 

/ vx  n 

or  in  words,  the  pressure  volume  product  of  a given  mass  of 
any  gas  is  proportional  to  its  absolute  temperature.  If  we 

differentiate  this  equation  with  respect  to  T and  divide  the 
result  by  the  original  equation,  we  have 


1 d (pv)  _ 1 
pv  cl  T ~ T 


(17) 


or  stated  in  words,  the  pressure  volume  product  (or  the  pressure 
at  constant  volume  or  the  volume  at  constant  pressure)  of  any 
gas  increases  by  one  T th  part  of  itself  for  each  rise  of  one  degree 
in  its  temperature,  T being  the  initial  absolute  temperature  of 
the  gas;  or,  since  p , v and  T in  the  above  equation  may  have 
any  values,  this  equation  also  states  that  the  increase  in  the 
pressure-volume  product  of  a given  mass  of  any  gas  per  degree 
rise  in  temperature  is  equal  to  2i\-i  °f  its  value  at  0°  C. 
These  statements  comprise  what  is  frequently  referred  to  as  the 
law  of  Charles.® 

All  of  these  statements  are  evidently  much  more  elegantly 
and  concisely  expressed  by  equation  (11)  from  which  we  have 
deduced  them.  In  fact  one  of  the  great  advantages  of  the 
language  of  the  mathematician  is  the  clear,  concise,  complete 
and  exact  character  of  its  statements.  The  simple  statement, 
that 


d2(pz;) 

d£2 


(18) 

7 '-JO—  M Y'  ( I la 


conveys  to  the  mind  of  the  mathematician  nearly  everything 
which  we  have  employed  3 pages  in  explaining.  .Because  of 
these  manifest  advantages  of  the  language  and  methods  of 


“Jacques  Alexandre  Cesar  Charles  (1746-1823).  French  mathematician 
and  physicist,  Professor  of  Physics  at  the  Conservatoire  des  Arts  et  Metiers. 
He  was  the  first  to  emploj^  hydrogen  for  the  inflation  of  ballons.  In  1787 
he  anticipated  Gay  Lussac’s  law  of  the  expansion  of  gases. 


26 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


the  mathematician  we  shall  use  them  freely  throughout  this 
book  at  the  same  time  translating  them,  wherever  desirable, 
into  more  familiar  forms  of  statement.  It  is  hoped  that  this 
dual  method  of  treatment  will  aid  the  student  in  gaining  a clearer 
insight  into  the  general  laws  and  principles  of  his  science  as  well 
as  giving  him  a better  appreciation  of  the  value  of  higher  mathe- 
matics to  the  chemist  than  he  may  perhaps  have  obtained  in 
his  college  courses  in  mathematics. 

6.  Dalton’s  Law  of  Partial  Pressures. — In  a mixture  of  gases 
we  have  more  than  one  species  of  molecule  and  the  total  pressure 
exerted  by  the  mixture  upon  the  walls  of  the  containing  vessel 
may  be  considered  as  the  sum  of  all  the  separate  partial  pressures 
due  to  the  impacts  of  the  molecules  of  the  different  gases.  That  is 

P = Pa+Pb+Pc+ (19) 

By  employing  the  same  method  of  reasoning  used  in  section  2 
above  we  can  readily  show  that  these  partial  pressures  are  each 
expressed  by  an  equation  of  the  same  form  as  equation  (4),  thus 


VAV  = inAmAu2A 

(20) 

Pbv  — i^BmB  U*B 

(21) 

VcV  = incmcu2c,  etc., 

(22) 

or  stated  in  words:  In  a mixture  of  gases  each  gas  exerts  the 
same  pressure  as  it  would  exert  if  it  were  alone  present  in  the 
volume  occupied  by  the  mixture.  This  statement  is  known  as 
Dalton’s  Law  of  Partial  Pressures. 


Problem  1. — Show  that  the  partial  pressure,  p a,  of  any  constituent  A 
of  a gaseous  mixture  is  given  by  the  relation 

n a 

PA  = -P  (23) 

where  n a is  the  number  of  molecules  of  A and  n is  the  total  number  of  all 
molecules  present,  p being  the  total  pressure  of  the  mixture.  Show  also  that 
UA 

the  molecular  fraction,  > must  also  be  equal  to  the  number  of  gram  mo- 
71 

lecular  weights  of  the  constituent,  A,  divided  by  the  sum  of  the  numbers  of 
gram  molecular  weights  of  all  the  gases  present  in  the  mixture. 


Sec.  7] 


THE  GASEOUS  STATE  OF  AGGREGATION 


27 


This  latter  ratio  is  called  the  mole -fraction  of  the  constituent 
in  question  and  is  represented  by  the  letter,  x.  Equation  (23) 
may,  therefore,  be  written, 

Pa  = ZaP  (24) 

and  similarly  for  another  constituent  of  the  mixture 

Pb  = zbP  (25) 

and  so  on  for  each  gas  present  in  the  mixture.  Stated  in  words: 
The  partial  pressure  of  any  gas  in  a mixture  is  equal  to  its  mole- 
fraction  in  the  mixture  multiplicity  the  total  pressure  of  the 
mixture. 

If  a closed  palladium  v essSF*  or  tube  connected  with  a man- 
ometer be  evacuated  and  cnen  placed  in  a gaseous  mixture  at 
high  temperature  cont^kimg  hydrogen,  the  hydrogen  will  diffuse 
through  the  palla^ipm  wall  into  the  vessel  and  will  register  on 
the  manometers^bs  partial  pressure  in  the  mixture.  The  palla- 
dium is  imp«jQieable  to  the  other  gases  but  easily  permeable  to 
hydrogen  which  passes  through  it,  until  its  partial  pressures  on 
the  two  sides  of  the  wall  become  the  same.  This  device  gives 
us  a means  of  demonstrating  by  direct  experiment  and  of  measur- 
ing the  partial  pressure  exerted  by  the  molecules  of  one  gas  in 
a mixture  of  several  gases. 

7.  The  Equation  of  State  of  a Perfect  Gas. — The  laws  which  we 
have  just  derived,  known  as  the  perfect  gas  laws,  may  be  conveni- 
ently combined  into  a single  expression.  To  do  this  we  have  only 
to  modify  slightly  equation  (16)  above.  The  numerical  value  of 
the  proportionality  constant,  k,  of  this  expression  depends  both 
upon  the  mass  of  gas  taken  and  upon  the  nature  of  the  gas,  but 
if  we  agree  that  the  mass  taken  shall  always  be  one  gram  mo- 
lecular weight  (called  also  one  mole)  in  every  instance,  then  by 

vv 

Avogadro’s  law  (II,  4)  the  value  of  is  the  same  for  all  gases, 
that  is, 

(26) 

where  R is  a universal  constant  which  is  independent  of  the  nature 
of  the  gas  and  v0  is  by  our  agreement  the  volume  of  one  gram 
molecular  weight  or  one  mole  of  the  gas.  It  is  called  the  molal 


28 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


volume  of  the  gas.  If  we  wish  to  extend  this  equation  to  apply 
to  any  quantity  of  a gas,  we  have  only  to  multiply  both  sides  by 
N,  the  number  of  moles  taken,  and  we  have 

pv  — NRT  (27) 


where  v,  the  total  volume  occupied  by  the  N moles  of  gas,  is  written 
for  Nv o.  This  expression  is  the  equation  of  state  of  a perfect 
gas  or  more  briefly  the  perfect  gas  law.  In  this  equation  p,  v, 
and  T denote  respectively  the  pressure,  volume  and  absolute 
temperature  of  the  . gas  while  N,  the  number  of  moles  present,  is 
m 

evidently  equal  to  ~ where  m is  the  mass  of  the  gas  and  M its 


molecular  weight.  R is  a constant,  the  same  for  all  gases,  and 
vv 

equal  to  the  value  of  for  one  mole  of  any  perfect  gas.  Its 


numerical  value  obviously  depends  upon  the  units  in  which  pv 
is  expressed.  If  p is  expressed  in  atmospheres  and  v in  liters,  R 
has  the  value  0.08207  liter-atmospheres  per  degree;  if  p is  ex- 
pressed in  dynes  per  square  centimeter  and  v in  cubic  centi- 
meters, then  R has  the  value  8.3162  X107  ergs  or  8.3162  joules  per 
degree;  while  if  pv  is  expressed  in  calories,  then  R has  the  value 
1.9852  calories  per  degree. 

8.  Temperature  and  Molecular  Kinetic  Energy. — It  has  al- 
ready been  stated  (II,  1)  that  experiments  have  shown  that  the 
mean  kinetic  energy  of  the  molecules  of  a perfect  gas  is  dependent 
only  upon  the  temperature.  The  nature  of  the  dependence  can 
be  deduced  by  combining  equation  (4)  with  equation  (27)  so 
as  to  eliminate  pv.  This  gives 

i nmu2  = NRT  (28) 

or  = f — RT=  f = cT  (29) 

n “ IN 


where  N is  Avogadro’s  number  and  e is  evidently  a constant. 
Stated  in  words : The  mean  kinetic  energy  of  the  molecules  of  a 
perfect  gas  is  proportional  to  the  absolute  temperature  of  the  gas 


and  the  proportionality  constant*  is  equal  to 


3 R 

2 N 


Problem  2. — From  the  value  of  R and  the  atomic  weights  required  calcu- 
late the  molecular  velocity,  u,  in  miles  per  second  for  the  following  gases: 


Sec.  9] 


THE  GASEOUS  STATE  OF  AGGREGATION 


29 


H2  atO°;  H2  at  3000°;  02  at  0°;  CeHe  at  0°;  Hg  at  0°.  (First  eliminate  N 
andm  from  equation  (29)  by  introducing  the  molecular  weight,  M,  of  the 
gas.) 

9.  The  Validity  of  the  Perfect  Gas  Laws. — It  will  be  remem- 
bered that  the  derivation  given  above  for  the  perfect  gas  law  is 
based  upon  equation  (4)  which  was  itself  derived  on  the  assump- 
tions (1)  that  the  molecules  of  a gas  are  so  far  apart  that  they 
exert  no  attraction  upon  one  another;  and  (2)  that  the  space 
which  they  themselves  actually  occupy  is  negligibly  small  in 
comparison  with  the  volume  of  the  containing  vessel.  No  real 
gas  exactly  fulfills  either  of  these  conditions,  but  it  is  evident 
that  all  gases  should  approach  these  conditions  more  closely  the 
lower  the  pressure,  that  is,  the  farther  apart  the  molecules  become. 
The  perfect  gas  law,  therefore,  is,  strictly  speaking,  only  a limit- 
ing law  which  may  be  considered  as  holding  exactly  only  for  an 
imaginary  gas,  called  a perfect  gas,  but  which  all  real  gases  should 
obey  more  and  more  closely  the  lower  the  pressure.  Experi- 
ment shows  that  this  is  actually  the  case.  The  magnitude  of 
the  divergence  of  real  gases  from  the  requirements  of  the  perfect 
gas  law  varies  with  the  nature  of  the  gas  and  its  temperature, 
but  for  pressures  not  greatly  in  excess  of  one  atmosphere  the 
divergence  is  small  (less  than  1 per  cent.)  for  gases  at  tempera- 
tures far  removed  from  their  maximum  condensation  tempera- 
ture. This  is  illustrated  by  the  data  in  the  first  half  of  Tables 
II  and  III.  The  figures  given  in  the  third  column  are  the  critical 
temperatures  of  the  gases.  The  critical  temperature,  or  the 
maximum  condensation  temperature  of  a gas,  is  the  highest  tem- 
perature at  which  the  gas  can  be  liquefied  by  increase  of  pressure. 

At  higher  pressures  and  at  lower  temperatures  all  gases  deviate 
more  and  more  from  the  requirements  of  the  perfect  gas  law. 
The  behavior  of  carbon  dioxide  (tc  = 31.35°)  toward  Boyle’s 
law  at  various  temperatures  is  shown  graphically  in  Figs.  1 and 
2.  The  way  in  which  the  py-product  varies  with  the  pressure 
at  the  temperatures  indicated  is  shown  by  the  curves.  Fig.  2 is 
on  a larger  scale  than  Fig.  1 . The  general  behavior  of  carbon  diox- 
ide as  indicated  by  these  diagrams  is  characteristic  of  all  gases. 

Problem  3. — Discuss  the  relations  displayed  graphically  in  Figs.  1 and  2, 
stating  in  words  all  the  relations  represented  by  these  curves. 


30 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


Fig.  1. 


Fig.  2. 

Reproduced  by  permission  of  Longmans  Green  & Co. 
(From  Young’s  Stoichiometry.) 


Sec.  9] 


THE  GASEOUS  STATE  OF  AGGREGATION 


31 


Table  II 


Values  of  for  various  gases.  Illustrating  the  magnitude  of  the  devia- 
tions of  gases  from  the  perfect  gas  law  and  from  Berthelot’s  equation  of 
state.  For  a perfect  gas  = 1- 


Gas 

Mol.  wt., 
M 

Critical 

tempera- 

ture, 

tc 

pv  0 

Observed  values  of  ^ j,  and  per 

cent,  deviations  from  unity. 

Values  of  calculated 

K 1 

from  Berthelot’s  Equation 
and  per  cent  deviations 
from  observed  values. 

t = 

0° 

< = 0° 

p = 0.1  atmos- 
phere 

p = 1 atmos- 
phere 

p = 1 atmosphere 

% 

% 

He 

4.00 

-268° 

1.000 

0.00 

1.0001 

0.01 

1.0006 

-0.05 

H2 

2.0155 

-242 

1.0002 

0.02 

1.0007 

0.07 

1.0005 

-0.02 

n2 

28.016 

-146 

0.9999 

0.01 

0.9995 

0.05 

0.9997 

+ 0.02 

02 

32.0000 

-119 

1.0000 

0.00 

0.9992 

0.08 

0.9992 

0.00 

A 

39.88 

-117 

0.9999 

0.01 

0.9992 

0.08 

0.9993 

0.01 

CH4..... 

16.013 

- 82 

0.9979 

0.21 

0.9982 

0.03 

Kr 

82.92 

— 62 

0.9978 

0.2 

0.9974 

0.04 

Xe 

130.2 

14.7 

0.9931 

0.7 

0.9926 

0.05 

C02 

44.00 

31 

0.9993 

0.07 

0.9932 

0.68 

0.9931 

0.01 

N20 

44.016 

36 

0.9998 

0.02 

0.9931 

0.69 

0.9925 

0.06 

HC1 

36.4678 

52 

0.9992 

0.08 

0.9926 

0.74 

0.9925 

0.01 

NH3.  . . . 

17.0333 

132 

1.0003 

0.03 

0.9865 

1.35 

0.9883 

0.18 

S02 

64.07 

157 

0.9973 

0.27 

0.9759 

2.41 

0.9813 

0.54 

Table  III 

Values  of  —j,  f°r  various  saturated  vapors  at  the  normal  boiling  points, 

tB,  of  the  liquids.  Illustrating  the  deviations  of  saturated  vapors  from  the 
perfect  gas  law  and  from  Berthelot’s  equation  of  state.  For  a perfect  gas 
pv  o 1 

RT  A- 


Vapor 

Mol. 

wt., 

M 

Critical 

tem- 

perature, 

tc 

Boiling 

point, 

tB 

Observed 
value  of 
pro 

RT  at  <B° 
p = 1 atmos- 
phere 

Per  cent, 
dev. 
from 
unity 

Value  of 
pro 

RT 

from  Berthe- 
lot’s 

equation 

Per  cent. 

diflf.  ' 
between 
calc, 
and 
obs. 
values 

He 

4.00 

-268 

— 268.5 

0.92 

8 

N2 

28.016 

-146 

-195.7 

0.95 

5 



0.95 

0 

02 

32.000 

-119 

-182.9 

0.97 

3 

0.970 

0 

S02 

64.1 

157 

- 10.0 

0.98 

2 

0.98 

0 

n— CsHi2.  . . . 

72.1 

197 

36.3 

0.956 

5 

0.958 

0.0 

n— C7H16.  . . . 

100.0 

267 

98.4 

0.948 

5 

0.955 

0.7 

CeHsF 

96.0 

287 

85.2 

0.974 

3 

0.967 

0.7 

C6H6 

78.1 

289 

80.1 

0.977 

2 

0.967 

1.0 

SnCU 

260.3 

318.7 

114.1 

0.964 

4 

0.964 

0 

CsHsCl 

112.5 

359 

132.0 

0.948 

5 

0.966 

1.9 

32 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


10.  The  Equations  of  van  der  Waals  and  of  Berthelot.  (a)  The 
Equation  of  van  der  Waals. — In  order  to  take  into  account  the 
attractive  forces  between  the  molecules  of  real  gases  as  well  as 
the  volume  which  the  molecules  themselves  actually  fill  and  thus 
to  obtain  an  equation  of  state  which  would  hold  more  accurately 
than  the  perfect  gas  law,  especially  lor  gases  at  high  pressures, 
van  der  Waals®  reasoned  as  follows:  Consider  a sphere  of  volume, 
Vq,  filled  with  one  mole  of  a gas  at  the  pressure,  p,  and  the  tem- 
perature, T.  Consider  the  layer  of  molecules  which  at  any  mo- 
ment are  just  about  to  strike  the  inner  surface  of  the  sphere.  The 
force  of  their  impacts  and  therefore  the  pressure  exerted  by  them 
will  be  diminished  owing  to  the  attractive  force  exerted  by  all  the 
other  molecules  of  the  gas  behind  them.  This  force  of  attrac- 
tion, /,  will  be  proportional  to  the  number  of  molecules  at  the 
surface  and  also  proportional  to  the  number  in  the  interior  of 
the  gas  and  each  of  these  in  turn  will  be  proportional  to  the 
density  of  the  gas,  or  inversely  proportional  to  the  volume  of 
the  sphere.  The  force  of  attraction  will,  therefore,  be  propor- 
tional to  the  square  of  the  density  or  inversely  proportional  to 
the  square  of  the  volume,  or  in  mathematical  language 


(30) 


where  a is  the  proportionality  constant  and  / is  expressed  in 
pressure  units.  Between  the  actual  pressure,  p,  exerted  by  the 
gas  molecules  and  the  “ perfect  gas  pressure/’  pp,  that  is,  the 
pressure  which  would  be  exerted  if  the  force  of  attraction  between 
the  molecules  were  absent,  there  will  evidently  exist  the  relation, 


(31) 


Again  the  actual  volume,  Vo,  of  the  sphere  is  not  the  volume  vp 
which  the  molecules  have  to  move  freely  about  in,  but  is  smaller 
than  this  by  an  amount,  b,  a quantity  which  is  a function2  of  the 
total  volume  occupied  by  the  molecules  themselves.  We  have 
therefore, 


vp=(v0  — b) 


(32) 


“ Joannes  Diderik  van  der  Waals,  Professor  of  Theoretical  Physics  at 
the  University  of  Amsterdam. 


Sec.  10]  THE  GASEOUS  STATE  OF  AGGREGATION 


33 


Now  on  substituting  this  corrected  pressure  and  volume  in 
equation  (26)  derived  above , we  have 

(p+,%)(v0-b)  = RT  (33) 


which  is  van  der  Waals’  equation  of  state  for  one  mole  of  a gas, 

a and  b being  constants  characteristic  of  the  gas  in  question. 

Experiments  show  that  the  general  behavior  of  gases  as  dis- 
played graphically  by  Figs.  1 and  2 is  very  well  expressed  mathe- 
matically by  the  equation  of  van  der  Waals.  But  although  the 
equation  of  van  der  Waals  correctly  describes  the  general  be- 
havior of  gases  (and  likewise  of  liquids)  throughout  the  whole 
range  of  pressures  and  temperatures,  it  fails  to  represent  quantita- 
tively the  actual  experimental  data  in  many  instances.  Various 
modifications  (over  100  in  all)  of  the  equation  have  been  pro- 
posed from  time  to  time  by  different  physicists  with  the  purpose 
of  diminishing  this  objection.  The  only  one  of  these  which  we 
shall  consider  here  is  that  of  Daniel  Berthelot.® 

(b)  Berthelot’s  Equation. — The  characteristic  constants,  a and 
b,  of  van  der  Waals’  equation  have  been  found  to  bear  a definite  re- 
lation to  the  critical  constants  of  the  gas,  from  which  in  fact 
they  may  be  computed.  With  the  aid  of  this  empirical  relation 
Berthelot  modified  the  equation  of  van  der  Waals  by  sub- 
stituting in  place  of  a and  b their  values  expressed  in  terms  of 
the  critical  constants  of  the  gas.  The  resulting  equation  after 
an  algebraic  rearrangement  may  be  expressed  as  follows: 


or 


=RT  [>+  iL  l r 
w=nrt  |h- 


(34) 

(35) 


where  pc,  the  critical  pressure,  is  the  pressure  required  to  con- 
dense the  gas  at  its  critical  temperature,  Tc,  and  the  other  quan- 
tities have  the  significance  previously  given  to  them.  The 
numerical  constants  are  empirical  ones.  It  is  evident  that  this 
expression  approaches  the  perfect  gas  law  as  p decreases  or  T 
increases. 


a Daniel  Berthelot  (1865-  ) Professor  of  Physics  in  the  University  of 

Paris. 


34 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


The  equation  of  Berthelot  not  only  expresses  the  general  be- 
havior of  all  pure  gases  thus  far  investigated  but  it  does  so  with 
a high  degree  of  accuracy,  even  for  comparatively  high  pressures 
in  some  instances.  This  is  well  illustrated  by  problems  4,  5,  and 
6,  below  and  also  by  the  data  shown  in  Table  II  where  the  values 


of 


pv o 

RT 


calculated  by  means  of  this  equation  agree  within  the  ex- 


perimental errors  with  the  values  found  by  direct  measurement. 
Even  in  the  case  of  saturated  vapors  the  results  of  experiment 
can  be  fairly  well  represented  by  Berthelot’s  equation,  as  shown 
by  the  data  in  Table  III. 

In  using  the  Berthelot  equation  of  state  it  should  be  remem- 
bered that  it  is  applicable  only  to  a pure  gas,  that  is,  to  a gas  in 
which  at  all  temperatures  and  pressures  up  to  the  critical  tem- 
perature and  pressure,  there  is  only  one  species  of  molecule. 
Moreover,  in  deriving  his  equation  Berthelot  explicitly  assumes 
that  the  gas  is  not  at  or  near  its  critical  temperature  and  pres- 
sure and  the  equation  is,  therefore,  only  applicable  to  cases  where 
this  condition  is  also  fulfilled.  In  other  words  the  equation  must 
not  be  applied  to  mixtures  of  gases,  or  to  partially  associated  or 
dissociated  gases,  or  to  gases  in  the  neighborhood  of  the  critical 
point.  Berthelot’s  equation  had  found  its  most  important  appli- 
cation in  the  exact  determination  of  molecular  and  atomic 
weights  as  will  be  further  explained  in  a following  section. 


Problem  4.— The  measured  value  of  for  hydrogen  at  0°is  1.006  for 

p = 10  atmospheres  and  1.032  for  p = 50  atmospheres.  Compute  the  values 
by  means  of  Berthelot’s  equation  of  state  and  compare  with  the  measured 
values.  (See  Table  XII  for  data  required.) 

Problem  5. — Make  similar  calculations  for  hydrogen  at  100°.  The  ob- 
served values  are  1.005  for  10  atmos.  and  1.025  for  50  atmos. 

Problem  6. — Make  similar  calculations  for  hydrogen  at  —104°.  The 
observed  values  are  1.007  at  10  atmos.  and  1.039  at  50  atmos. 

Problem  7. — Make  a similar  calculation  for  CO2  at  0°  and  50  atmos. 
The  observed  value  is  0.098.  Note  that  0°  and  50  atmospheres  are  close 
to  the  critical  values  for  this  gas  and  that  Berthelot’s  equation  cannot  be 
employed  under  these  conditions  as  the  result  of  your  calculation  indicates. 


11.  The  Boyle  Point. — From  the  observed  values  of  given 
in  Table  1 1 it  is  evident  that  at  0°  the  deviation  from  unity  is 


Sec.  12]  THE  GASEOUS  STATE  OF  AGGREGATION 


35 


positive  in  the  case  of  hydrogen  and  helium  and  negative  in  the 
case  of  all  the  other  gases.  At  lower  temperatures,  however,  the 
deviation  becomes  negative  in  the  case  of  hydrogen  also,  while 
at  higher  temperatures  the  deviation  in  the  case  of  the  other 
gases  changes  from  negative  to  positive.  In  other  words,  there 
is  for  each  gas  a certain  temperature,  called  the?  Boyle  point,  at 
which  the  gas  obeys  the  laws  of  Boyle  and  Avogadro  exactly. 
Above  this  temperature  the  deviation  from  this  law  is  positive 
while  below  this  temperature  it  is  negative. 


Problem  8. — By  means  of  Berthelot’s  equation  of  state  show  that  the 
Boyle  point,  Tj , of  a gas  is  connected  with  its  critical  temperature  by  the  re- 
lation, Ti  — 2.44  Tc.  Compute  the  Boyle  point  for  H2,  I2,  N2,  C02,  CH4 
and  NH3. 

12.  Molecular  Weights  and  Densities  of  Gases,  (a)  Approxi- 

7YI 

mate  Molecular  Weights  from  Gas  Densities. — Since  N = ^ and 

771 

by  definition  D = — , the  perfect  gas  law  may  evidently  be  written, 


mRT=DRT 
pv  p 


(36) 


from  which  the  molecular  weight  of  a gas  can  be  calculated  if  its 
density,  D,  is  known  at  some  temperature,  T,  and  pressure,  p. 
It  is  frequently  customary  to  refer  the  densities  of  gases  to  that 
of  some  gas  employed  as  a standard.  Thus  the  density  referred 
to  oxygen  for  any  gas  signifies  the  ratio  of  its  density,  D,  to  that 
of  oxygen,  D0,  at  the  same  temperature  and  pressure.  By 
dividing  equation  (36)  by  the  corresponding  equation  for  oxygen 

/ Dn  RT\ 

( i.e.,  32  = ) we  have 

V V / D 

M = 32  yr  (37) 

uo 


or  in  words,  the  molecular  weight  of  any  gas  is  equal  to  32  times 
its  density  referred  to  oxygen.  Similarly  if  we  choose  to  employ 
air  as  the  reference  gas,  the  above  equation  would  read 

D 

M = 28.97^ 


(38) 


36 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


where  DA  is  the  density  of  air  at  the  same  temperature  and  pres- 
sure and  28.97,  the  so-called  “ molecular  weight”  of  air,  is  simply 
the  value  which  experiment  gives  for  the  quantity,  M,  in  equa- 
tion (36)  when  experimental  data  for  air  are  substituted  on  the 
right-hand  side  of  this  equation: 

Since  real  gases  do  not  obey  the  perfect  gas  law  exactly,  except 
in  the  limiting  case  of  very  low  pressures,  equation  (37)  will  give 
accurate  values  for  the  molecular  weight  of  a gas  only  if  we 


employ  for^  the  value  approached  by  this  ratio  as  the  pressure 

approaches  zero.  It  is  quite  possible,  as  we  shall  see  in  the 
second  part  of  this  section,  to  determine  this  limiting  value  with 
a hi  gh  degree  of  accuracy  so  that  this  equation  can  be  and  is  em- 
ployed for  the  accurate  determination  of  molecular  and  atomic 
weights.  In  the  case  of  most  gases,  however,  it  is  sufficient 
to  determine  the  molecular  weight  approximately  by  means  of 
equation  (36)  or  (37)  for  the  exact  molecular  weight  can  then  be 
obtained  simply  by  multiplying  the  combining  weight  of  the 
substance  by  the  small  whole  number  (or  whole  number  ratio) 
which  gives  a product  most  nearly  equal  to  the  approximate 
molecular  weight. 


Problem  9. — Calculate  the  approximate  density  of  oxygen  under  standard 
conditions,  that  is,  at  0°  C.  and  a pressure  of  one  atmosphere.  (Its  actually 
measured  density  under  standard  conditions  is  1.4292  grams  per  liter.) 
Under  the  same  conditions  nitrogen  weighs  1.2514  grams  per  liter.  What  is 
its  density  referred  to  oxygen?  From  the  above  data  calculate  the  approxi- 
mate value  of  its  molecular  weight.  The  analysis  by  Guye  and  Drouginin 
of  one  of  the  oxides  of  nitrogen  gave  the  composition  1 part  of  oxygen  and 
0.43781  part  of  nitrogen.  What  is  the  combining  weight  of  nitrogen  in  this 
compound?  From  this  result  and  the  approximate  value  for  the  molecular 
weight  obtained  above  calculate  a more  exact  value  for  the  molecular  weight 
of  nitrogen. 


> 


(b)  Exact  Molecular  Weights  from  Gas  Densities.3 — If  we 

apply  the  reasoning  of  the  preceding  paragraphs  to  Berthelot’s 
equation  instead  of  to  the  perfect  gas  law,  it  will  be  easily  seen 
that  we  shall  obtain,  in  place  of  equation  (36),  the  more  exact 
relation, 


M = 


DRT 

V 


1 + 


9Tc(T*-GT>)\  \ 

128  p T3  iPJ 


(39) 


Sec.  12]  THE  GASEOUS  STATE  OF  AGGREGATION 


37 


which  for  the  sake  of  brevity  may  be  written 

M = I~(l+Ap)  (40) 

where  A is  written  in  place  of  the  expression  in  the  braces.  From 
this  relation  the  molecular  weight  of  a gas  can  be  very  accurately 
calculated  as  will  be  understood  from  the  solution  of  the  following 
problems : 

Problem  10. — From  the  data  given  in  Table  XII  calculate  exact  values 
for  the  molecular  weights  of  O2,  N2,  Ar,  N20  and  CH4,  with  the  aid  of 
Berthelot’s  equation  of  state. 

Problem  11. — At  0°  and  0.5  atmosphere  the  density  of  neon  is  0.44986 
and  that  of  sulphur  dioxide  is  1.4807  grams  per  liter.  From  these  values 
and  the  densities  under  standard  conditions  (Table  XII)  compute  the  exact 
atomic  weights  of  neon  and  of  sulphur,  without  making  any  use  of  the  crit- 
ical data  in  either  instance. 


Problem  11  shows  that  BertheloFs  method  can  be  employed 
to  determine  molecular  weights  accurately  without  the  necessity 
of  knowing  the  critical  constants.  Measurement  of  the  density 
of  the  gas  at  two  different  pressures  is  all  that  is  required.  This 
method  of  making  the  calculation  is  usually  known  as  the  method 
of  limiting  densities  and  is  in  general  more  reliable  than  the 
method  of  critical  constants,  illustrated  by  problem  10,  because 
accurate  values  for  the  critical  constants  are  rather  difficult  to 
obtain. 

If  as  above  in  our  treatment  of  the  perfect  gas  equation  we 
choose  to  make  use  of  densities  referred  to  oxygen  instead  of 
absolute  densities,  we  have  only  to  apply  the  same  reasoning, 
that  is,  we  divide  equation  (40)  by  the  corresponding  equation 
for  oxygen, 

32  = D°pT  (1+Aop)  (41) 


and  thus  obtain  in  place  of  equation  (37)  the  more  exact  relation, 


M = 32 


D 

Do 


l+Ap 
1 -\-Aop 


(42) 


which,  since  at  moderate  pressures  Ap  and  A op  are  small  in 
comparison  with  unity,  may  also  be  written 

M = 32^(1  + (A  -A0  )p)  = 32^(1  — ap) 


(43) 


38 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


Problem  12. — Under  standard  conditions  N20  is  1.38450  times  as  dense 
as  oxygen.  Calculate  its  molecular  weight  by  the  method  of  critical  con- 
stants employing  equation  (43)  and  taking  the  necessary  critical  data  from 
Table  XII. 

Problem  13. — Under  standard  conditions  the  density  of  argon  referred 
to  oxygen  is  1.24482.  At  the  same  temperature  but  at  a pressure  of  0.5 
atmosphere  it  is  1.24626.  Calculate  the  molecular  weight  of  argon  by 
the  method  of  limiting  densities  employing  equation  (43). 

Problem  14. — At  0°  and  a pressure  of  0.5  atmosphere  the  density  of  oxy- 
gen is  0.71485  gram  per  liter  and  at  1 atmosphere  it  is  1.4290  grams  per  liter. 
From  these  two  data  calculate  the  numerical  value  of  the  gas  constant,  R. 

Berthelot’s  methods  of  calculating  exact  molecular  weights 
from  gas  densities  are  of  particular  value  in  connection  with  the 
inert  gases  of  the  argon  group  for  since  these  gases  have  no  com- 
bining weights  their  molecular  and  atomic  weights  cannot  be 
accurately  obtained  by  the  ordinary  method.  In  Table  IV 
below,  atomic  weights  obtained  from  gas  densities  are  compared 
with  those  obtained  from  the  combining  weights  for  four  elements. 
The  agreement  is  within  the  experimental  errors  in  every  instance. 

Table  IV. — Comparison  of  Atomic  Weights  Determined  from  Gas 
Densities  with  those  Determined  From  Combining  Weights 

0 = 16.000 

(Four  Elements  for  which  the  most  accurate  data  of  both  kinds  are  avail- 
able.) (The  values  given  are  averages.) 


Element 

H 

C 

N 

Cl 

From  gas  density 

1.00775  | 

12 . 004 

14.008 

35.461 

From  combining  weight. . . 

1.00775 

12.00 

14.008 

35 . 460 

PROBLEMS 

(Assume  the  perfect  gas  laws.) 

Problem  15.— What  is  the  volume  of  100  grams  of  ether  vapor  (C4H10O) 
at  10°  and  60  cm.?  What  is  its  absolute  density  and  its  density  referred  to 
oxygen  under  the  same  conditions? 

Problem  16. — What  would  be  the  volume  of  a mixture  of  1 gram  of  carbon 
dioxide  and  4 grams  of  carbon  monoxide  at  17°  and  a total  pressure  of  0.1 
atmosphere?  What  would  be  the  partial  pressure  of  each  gas? 

Problem  17. — A mixture  of  N2  and  Ar  at  371°  and  a pressure  of  671  mm. 
contains  25  per  cent,  of  N2  by  weight.  Calculate  (1)  the  partial  pressure  of 
each  gas,  (2)  the  absolute  density  of  the  mixture  at  371°  and  671  mm.,  and 
(3)  the  number  of  molecules  (not  moles)  of  each  gas  in  1 cubic  millimeter 
of  the  mixture. 


Sec.  12]  THE  GASEOUS  STATE  OF  AGGREGATION 


39 


Problem  18. — A balloon  open  at  the  bottom  and  filled  with  hydrogen 
occupies  a volume  of  2000  cubic  meters.  Calculate  in  tons  its  lifting  capacity 
at  sea  level  and  27°. 

Problem  19. — A Bessemer  converter  is  charged  with  1000  kg's,  of  iron 
containing  3 per  cent,  of  carbon.  How  many  cubic  meters  of  air  (containing 
25  per  cent.,  by  weight,  of  oxygen)  at  27°  and  1 atm.  are  needed  for  the 
combustion  of  all  the  carbon,  assuming  1/3  to  burn  to  C02  and  2/3  to 
CO?  What  will  be  the  partial  pressures  of  all  of  the  gases  evolved  by  the 
converter? 


REFERENCES 

Books:  (1)  Stoichiometry.  Sydney  Young,  1908.  Chapters  II,  III, 
X and  XI. 

Journal  Articles:  (2)  T.  W.  Richards,  Jour.  Amer.  Chem.  Soc.,  36, 
617  (1914).  (3)  Numerous  papers  in  the  Jour.  Chim.  Phys.  in  recent  years. 

See  also  Grinnell  Jones,  Jour.  Amer.  Chem.  Soc.,  32,  514  (1910). 


CHAPTER  III 


THE  LIQUID  STATE  OF  AGGREGATION 

1.  Liquefaction  of  a Gas  or  Vapor. — Consider  any  pure  sub- 
stance in  the  gaseous  state  enclosed  in  a transparent  cylinder 
provided  with  a movable  piston  (Fig.  3).  Let  the  cylinder  be 
surrounded  by  some  suitable  bath  by  means  of  which  its  tem- 
perature can  be  kept  constant  at  some  point  which  may  be  any- 
where between  the  critical  temperature  and  the  melting  point 
of  the  substance.  Now  let  the  pressure  upon  the  gas  be  gradu- 
ally increased  by  forcing  down  the  piston.  The  volume  of  the 
gas  will  be  observed  to  gradually  decrease  as  the  pressure  rises 
and  after  the  pressure  has  been  increased  to  a certain  value  (de- 
pending upon  the  nature  of  the  substance  and  the  temperature), 
a second  phase,  the  liquid  phase,  will  begin  to  appear.  This 
phase  usually  appears  first  in  the  form  of  a fine  mist  which  gradu- 
ally settles  to  the  bottom  of  the  cylinder  where  it  collects  to  a 
mass  of  liquid  which  is  seen  to  be  separated  from  the  gaseous 
phase  remaining  in  the  upper  part  of  the  cylinder  by  a sharp 
bounding  surface  (Fig.  4).  As  soon  as  the  liquid  phase  has  ap- 
peared in  the  system  no  further  increase  of  pressure  can  be  pro- 
duced as  long  as  any  of  the  gaseous  phase  remains.  Any  at- 
tempt to  increase  the  pressure,  ever  so  slightly,  will  result  in  the 
complete  condensation  (liquefaction)  of  all  of  the  gas  (Fig.  5). 
This  constant  pressure,  which  always  prevails  in  every  system 
composed  of  a gas  and  a liquid  in  equilibrium  with  each  other  at 
a given  temperature  is  calledthe  vapor  pressure  or  vapor  tension 
of  the  liquid  at  that  temperature  and  is  a characteristic  property 
of  the  liquid.  When  all  the  gas  has  been  liquefied  (Fig.  5)  the 
pressure  on  the  liquid  can  then  be  increased  indefinitely.  When 
the  pressure  on  a liquid  is  increased,  however,  the  corresponding 
decrease  in  volume  is  ordinarily  very  much  smaller  than  is  the 
case  with  a gas.  In  other  words  the  compressibility  of  liquids  is 
comparatively  small. 


40 


Sec.  2] 


THE  LIQUID  STATE  OF  AGGREGATION 


41 


If  we  were  to  carry  out  the  reverse  of  the  above  process,  start- 
ing with  a liquid  under  high  pressure  and  gradually  decreasing 
the  pressure,  we  should  observe,  after  the  pressure  had  reached 
a value  equal  to  the  vapor  tension  of  the  liquid,  that  some  of 
the  liquid  would  evaporate  and  the  system  would  again  become 
heterogeneous  owing  to  the  appearance  of  the  gaseous  phase. 


Fig.  3.  Fig.  4.  Fig.  5. 


This  behavior  serves  as  a definite  distinction  between  a liquid 
and  a gas.  (Cf.  II,  1.) 

2.  General  Characteristics  of  the  Liquid  State.  Surface  Ten- 
sion.— The  molecular  condition  in  the  interior  of  a body  of 
liquid  differs  from  that  within  the  body  of  a gas  in  degree  rather 
than  in  kind.  The  molecules  of  a liquid  possess  the  same  un- 
ordered or  random  motion  characteristic  of  gas  molecules  and 
experiments  show  that  in  a given  liquid  the  mean  kinetic  energy, 
%mu2,  of  the  moving  molecules  depends  only  upon  the  tempera- 


42 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  Ill 


ture,  being  independent  of  the  mass  of  the  molecules  and  the 
same  for  all  molecules  within  a given  liquid  phase.  The  mole- 
cules of  a liquid  are,  however,  much  closer  together  than  those 
of  a gas  under  low  pressure  and  hence  the  mean  free  path 
(II,  1)  of  a liquid  molecule  is  very  short  compared  with  that  of 
a gaseous  molecule.  Moreover,  the  close  proximity  of  the 
molecules  to  one  another  makes  the  effects  of  their  mutual  at- 
tractions very  pronounced  and  gives  rise  to  the  phenomenon  of 
surface  tension.  Consider,  for  example,  any  molecule  within 
the  body  of  a liquid.  It  is  very  powerfully  attracted  by  the  other 
molecules  which  closely  surround  it  but  since  it  is  surrounded 
on  all  sides  by  them  it  is  on  the  average  attracted  to  a like  de- 
gree in  all  directions  so  that  the  attractive  forces  balance  one 
another.1  The  molecules  in  the  surface  layer  of  the  liquid,  how- 
ever, are  attracted  only  downward  and  sideways,  not  upward, 
since  there  are  no  liquid  molecules  above  them.  As  a result 
of  these  attractions  the  surface  molecules  act  as  though  they 
formed  a tightly  stretched  but  elastic  “skin”  over  the  surface 
of  the  liquid  and  as  a result  of  this  surface  tension,  as  it  is  called, 
every  liquid  when  freed  from  the  influence  of  external  forces 
(such  as  gravitation,  for  example)  always  assumes  a spherical 
shape  since  of  all  possible  shapes  the  sphere  is  the  one  having 
the  smallest  surface.  The  surface  tension  always  acts  so  as 
to  make  the  total  surface  of  the  liquid  as  small  as  possible. 

A liquid  in  a vessel  under  the  influence  of  the  attraction  of  the 
earth  assumes  the  shape  of  the  vessel  and  has  a level  surface. 
This  is  because  the  earth’s  attraction  largely  overcomes  the  effect 
of  the  surface  tension.  At  the  point  of  contact  between  the 
surface  of  the  liquid  and  the  wall  of  the  vessel,  however, 
the  surface  is  not  level  but  always  bends  either  upward  or  down- 
ward according  to  whether  the  liquid  “wets”  or  does  not  “wet” 
the  wall  of  the  vessel.  Thus  water  in  glass  always  gives  a 
meniscus  which  is  concave  upward  while  mercury  which  does 

1 This  attractive  force  is  the  same  as  that  which  is  present  in  gases  under 

a 

high  pressure  and  which  was  represented  by  the  quantity,  , in  van  der 

Waals’  equation  (II,  10).  In  fact  the  equation  of  van  der  Waals  will  express 
very  closely  the  variation  of  the  volume  of  a liquid  with  changes  in  its  pres- 
sure and  temperature. 


Sec.  3] 


THE  LIQUID  STATE  OF  AGGREGATION 


43 


not  wet  glass  gives  a meniscus  which  is  concave  downward. 
This  effect  of  surface  tension  which  is  seen  at  the  boundary  be- 
tween the  surface  of  the  liquid  and  the  enclosing  wall  is  much 
more  pronounced  in  the  case  of  a liquid  surface  enclosed  in  a 
small  tube  or  pore.  If,  for  example,  a capillary  tube  of  glass  is 
dipped  into  water,  the  water  inside  the  capillary' rises  above  the 
level  of  that  outside  (Fig.  6),  while  if  such  a tube  is  dipped  into 
mercury,  the  level  in  the  tube  is  depressed  below  the  level  out- 
side. (Fig.  7.)  In  each  case  the  result  is  a smaller  total  surface 
for  the  liquid.  By  total  surface  in  this  case  is  meant  the  surface 
of  contact  between  the  liquid  and  some  other  phase  which  it 


Fig.  6.  Fig.  7. 

does  not  wet.  The  phenomenon  of  the  rise  (or  depression)  of 
liquids  in  capillaries  or  small  pores  of  any  description  is  known 
as  capillarity  and  is  due  to  the  surface  tension  of  the  liquid. 
Many  other  familiar  phenomena  such  as  the  behavior  of  soap- 
bubble  films,  and  the  absorption  of  water  by  a sponge,  are  the 
result  of  surface  tension. 

3.  The  Measurement  of  Surface  Tension. — The  unit  of  sur- 
face force,  which  is  also  called  the  surface  tension,  and  is  "repre- 
sented by  the  Greek  letter  gamma,  y,  is  defined  as  the  force 
which  acts  upon  a unit  line,  located  within  the  surface.  For 


44 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  Ill 


example,  if  an  attempt  were  made  to  increase  the  surface  of  a 
liquid  by  pulling  upon  an  imaginary  line  1 cm.  long,  located 
within  the  surface,  the  force  tending  to  oppose  the  increase  in 
surface  is  called  the  surface  tension,  7,  of  the  liquid.  The  mag- 
nitude of  the  surface  tension  is  a characteristic  property  of  the 
liquid.  Of  the  various  methods  which  are  employed  in  its 
measurement  we  shall  consider  here  only  two  of  the  most  im- 
portant ones. 

(a)  The  Capillary  Tube  Method. — When  a liquid  rises  (or  sinks) 
in  a capillary  tube  of  radius,  r,  the  height,  h,  to  which  it  rises 
must  evidently  be  such  that  the  surface  force,  fs,  which  is  hold- 
ing it  up  is  just  balanced  by  the  force  of  gravity,  fg,  which  is 
pulling  it  down,  that  is, 


/.-/»  (i) 

Now  the  surface  force  is  acting  upon  a line  of  length,  l,  which 
is  equal  to  the  inner  circumference  of  the  capillary  tube  and  hence 

fs  = h = 2irry  (2) 

The  force  of  gravity  acting  upon  the  volume,  V,  ( =irr2h ) of 
raised  liquid  is 


f0  = mg  = VDg  = irr2hDg  (3) 

where  m is  the  mass  and  D the  density  of  the  liquid  and  g is 
the  acceleration  due  to  gravity.  Hence 

fs=fg  = 2irry  = irr2hDg  (4) 

and  the  surface  tension, 

7 = \grhD  (5) 

Problem  1. — The  surface  tension  of  water  at  0°  is  73.21  dynes  per  centi- 
meter. How  high  will  water  at  this  temperature  rise  in  a glass  capillary 
0.1  mm  in  diameter? 

(b)  The  Drop-weight  Method.— When  a drop  of  liquid  forms 
slowly  at  a capillary  tip  (the  tip  of  a pipette,  for  example),  the 
size  attained  by  the  drop  just  before  it  breaks  away  from  the  tip 
will  evidently  depend  upon  the  surface  tension  of  the  liquid. 


Sec.  4] 


THE  LIQUID  STATE  OF  AGGREGATION 


45 


Lohnstein2  has  shown  that  for  a properly  constructed  tip  the 
weight,  w,  of  the  drop  which  falls  from  it  under  the  above  condi- 
tions is  given  by  the  expression, 


w 


= W(  vW  5 


(6) 


where  r is  the  internal  radius  of  the  tip  and  D is  the  density  of 
the  liquid.  For  a series  of  liquids  it  may  be  possible  to  choose 


a value  for  r such  that  the  function,  f 


.will  be  practically 


V2y  /D, 

constant  for  all  the  liquids  in  the  series  and  under  these  circum- 
stances the  above  expression  becomes,  approximately, 


w = ky 


(7) 


that  is,  the  drop -weights  of  these  liquids  from  this  particular 
tip  will  be  directly  proportional  to  their  surface  tensions.  Since 
drop-weights  can  usually  be  more  easily  and  accurately  measured 
than  the  capillary  rise,  the  drop-weight  method  is  especially 
valuable  for  determining  the  relative  surface  tensions  of  any 
series  of  liquids  which  resemble  one  another  somewhat.  In  order 
to  determine  the  absolute  value  of  the  surface  tension  by  the  drop- 
weight  method  the  capillary  tip  employed  should  be  standard- 
ized (i.e.,  the  value  of  k in  equation  (7)  should  be  determined) 
by  measurements  with  a liquid  whose  absolute  surface  tension 
is  known.  Using  the  drop-weight  method3  Morgan®  has  been 
able  to  determine  relative  surface  tensions  with  a precision  of 
a few  hundredths  of  1 per  cent. 

4.  The  Equations  of  Eotvos  and  of  Ramsay  and  Shields. — The 
smallest  surface  which  one  molal  weight  of  a liquid  can  have  is  its 
surface  when  in  the  form  of  a sphere.  This  surface,  which  is 
called  the  molal  surface,  and  is  represented  by  So,  is  evidently 
expressed  by  the  relation, 

i /M\  z 

S0*>Vo  z=a[D)3  (8) 


where  U0  is  the  molal  volume  of  the  liquid,  M its  molal  weight, 
and  D its  density.  The  product  of  the  surface  tension  into 

° J.  Livingston  R.  Morgan  (1872-  ).  Professor  of  Physical  Chemist^ 

at  Columbia  University,  New  York  City. 


46 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  Ill 


the  molal  surface,  ySo,  evidently  has  the  dimensions  of  work 
or  energy  and  is  called  the  molal  surface  energy.  It  represents 
the  work  involved  in  producing  the  surface,  So,  against  the 
surface  tension,  y.  It  is  analogous  to  the  molal  volume  energy, 
pvo,  in  the  case  of  a gas,  (II,  4). 

Between  the  molal  surface  energy  of  pure  (■ i.e .,  non-asso- 
ciated)  liquids  and  the  temperature  there  exists  a relation,  dis- 
covered by  Eotvos®  in  1886,  which  is  perfectly  analogous  to 
equation  (13,  II)  for  gases.  It  is  expressed  mathematically  by 
the  equation, 


— d(y$0) 
d t 


d£ 


const.,  °r 


IK 

- (\y\D 

~dt~ 


(9) 


where  Ks  is  a constant  which  has  the  same  value  for  all  pure 
liquids.  Expressed  in  words  this  equation  states  that:  The 

temperature  rate  of  change  of  the  molal  surface  energy  of 
a pure  liquid  is  independent  of  the  temperature  and  of  the 
nature  of  the  liquid.  The  independence  with  respect  to  the 
temperature  is  illustrated  by  the  data  given  in  Table  V. 


Table  V 

/My 

Illustrating  the  equation  of  Eotvos,  ~ dY  \P  / =Ka.  Values  of  Ks  for  ben- 
zene  at  various  temperatures  between  11°  and  120°.  tc=  288°. 


Measurements  by  Renard  and  Guye 

Measurements  by  Ramsay 

t°  = 

11.4 

31.2 

55.1 

68.5 

78.3 

80 

90 

100  110 

120 

200 

250 

Ks  = 

2.10 

2.13 

2.12 

2.10 

2 . lol 

2.09 

2.10 

2 . 10  2.10 

2.10 

2.10 

2.08 

The  integral  of  equation  (9)  is 
/HA* 

y{d)  = -Kst+K8K'-K8(K'-t)  (10) 

where  KSK'  is  the  integration  constant.  (Cf.  equation  14,  II.) 
Experiments  made  by  Ramsay6  and  Shields  using  a number  of 

° Baron  Roland  von  Eotvos  ( pr . Autvush),  Professor  of  Physics  in  the 
University  of  Budapest. 

b William  Ramsay,  K.  C.  B.,  F.  R.  S.  (1852-  ).  Professor  of 

Chemistry  (retired  1912)  at  University  College,  London. 


Sec.  4] 


THE  LIQUID  STATE  OF  AGGREGATION 


47 


different  liquids  showed  that  the  constant,  K' , has  approximately 
the  value, 

K'  = tc- 6 (11) 

where  tc  is  the  critical  temperature  of  the  substance  in  each  case. 
Equation  (10),  therefore,  becomes 

y(~y=K.(U-t- 6)  (12) 

which  is  known  as  the  Equation  of  Ramsay  and  Shields.  As  a 
rule,  this  relation  does  not  hold  unless  tc  — t is  greater  than  35°, 
that  is,  in  the  neighborhood  of  the  critical  temperature  the  rela- 
tion fails.  The  constant,  Ks,  has  the  average  value  2.1  ergs 
per  degree,  when  7 is  expressed  in  dynes.  This  value  is  based 
upon  the  assumption  that  the  substance  in  the  liquid  state  has 
the  same  molecular  weight  as  it  has  in  the  gaseous  state. 

Equation  (12)  may  also  be  written,  ySo  = kd,  where  0 repre- 
sents the  temperature  counted  downward  from  a point  6°  below 
the  critical  tempeiature  of  the  liquid.  This  form  of  writing 
the  equation  brings  out  its  resemblance  to  equation  (16,  II)  for 
gases. 

Table  VI 

Values  of  the  constant,  KSy  of  the  Eotvos  equation,  for  a variety  of  sub- 
stances calculated  on  the  assumption  that  the  molecular  weight  in  the 

liquid  state  is  the  same  as  that  in  the  gaseous  state.  Ks  = ^ £ —7  3 J 

Data  from  Walden’s  Tabulation  [Z.  physik.  Chem.,  82,  291  (1913)]. 


1.  Elements  and  Inorganic  Oxides  and  Halides 


Substance 

A | 

n2  1 

02  | 

Cl2 

1 

HC1  | HBr 

1 HI 

CO  | C02 

Mol.  wt 

39.9 

0 

00 

<N 

32 

71.92 

124 

36.46  80.97 

128.0 

28  l 44 

Ks  = 

2.0 

2.0 

1.9 

2.0 

2.2 

1.5(?)l  2.0 

2.0 

2.0  2.2 

Elements  and  Inorganic  Oxides  and  Halides. — Continued 


Substance 

j n2o  1 

SO3  | 

P4O6 

SOCI2 

| PCls  | 

SiCh 

CCU  | 

SnCU 

| CS2 

Mol.  wt 

. 

34.02 

80.06' 

220 

120.0 

137.4 

; 170.1 

153.8 

260.8 

76.14 

2.2 

2.3 

2.4 

2.1 

2.1 

2.0 

2.1 

2.2 

2.1 

2.  Hydrocarbons 


Substance 

i CeHe  I 

CeHsCHa 

n-C6Hi4  | 

C6H6C2H5 

m-CsHio 

Mol.  wt 1 

78.1 

92.06 

86.11 

106.1 

106.1 

1 2.1 

2.0 

2.1 

2.1 

2.2 

48 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  II 


Hydrocarbons. — Continued 


Substance 

CioHs  1 CsHie  l 

n-CsHis  | (C2H5)2  | 

(c2h5)2ch21 

CmHh 

Mol.  wt 

128.1  102.1 

114.1  58.08 

71.1 

182.1 

Ks  = 

2.3  2.3 

2.2 

2.2 

2.25 

2.5 

3.  Esters  and  Ethers 

Substance 

||  HC02CH3  | 

HC02C3H7  | CH3C02C2Hb  | C2H5C02CH3 

Mol.  wt 

I 60.03 

88.06 

88. 

06 

88.06 

Ks  = 

2.0 

2.1 

2. 

2 

2.2 

Esters  and 

Ethers. — Continued 

Substance 

HC02C5Hh  CChCOi-CsHii  II  C 

i2Hi602  C2oH3804 

C67HnoOe 

Mol.  wt 

116.1 

233.4 

192.1 

342.3 

890.9 

Ks  = 

2.1 

2.46 

2.6 

3.3 

5.7 

4.  Nitrogen  Compounds 

Substance 

n-C3H7NH2 

| C6H5NH2 

C6H5N(CH3)2  I 

CeHsNO? 

Mol.  wt 

. . . 59.08 

93.06 

121.1 

123 

Ks  = 

1 . 5(?) 

2.0 

2.3 

2.1 

Nitrogen  Compounds. — Continued 


Substance 

||  p-CeH40HN02 

| C9Hi203N2  1 

| CisH^Sb 

C2iH2iN 

Mol.  wt 

139 

196.1 

351.3 

287.2 

Ks  = 

1.8 

2.0 

3.5 

3.5 

5.  Hydroxyl  Compounds 


Substance 

H20 

CHsOH  | C2H5OH  1 

HC02H 

Mol.  wt 

18 

32.03 

46.1 

46 

1.1 

1.0 

1.2 

1.0 

Hydroxyl  Compounds. — Continued 


Substance 

| n-C3H7OH  | 

C3Hb(OH)3 

CeHsOH 

C22H4203 

Mol.  wt 

60.1 

92.06 

94.1 

354.3 

K,= 

1.2 

1.3 

1.7 

3.3 

In  Table  VI  are  shown  the  values  of  this  constant  for  a variety 
of  different  liquids.  The  data  given  in  the  first  four  sections  of 
this  table  show  that  the  value  of  this  constant  is  close  to  2.1  in 
the  case  of  nearly  all  of  the  liquids  given.  These  liquids  are 
evidently  of  the  most  varied  character  and  the  temperatures  are 
widely  different  in  the  different  cases,  ranging  from  — 183°  in 
the  case  of  oxygen  to  210°  in  the  case  of  diphenyl  methane, 


Sec.  5] 


THE  LIQUID  STATE  OF  AGGREGATION 


49 


(CeHs^CHo.  The  last  few  liquids  in  each  section  of  the  table, 
however,  have  values  of  K,  which  are  considerably  higher  than 
2.1  and  it  seems  to  be  a fairly  general  rule  that  substances  of 
very  high  molecular  weight  have  values  of  Ks  which  are  decid- 
edly larger  than  2.1.  In  fact  Walden®  and  Swinne  find4  that  this 
constant  can  be  approximately  calculated  from  the  expression, 

#,  = 1.90+0.011  (SnVZ)  (13) 

where  VA  is  the  sum  of  the  square  roots  of  the  atomic  weights 
of  the  elements  in  the  compound,  each  square  root  being  multi- 
plied by  the  corresponding  subscript  as  shown  by  the  formula 
of  the  compound.  Thus  for  tristearine,  C57H110O6,  we  have 
2raVA  = 57  Vl2+110Vl+6  Vl6  = 327  and  Ks=  1.90+0.011  X 
327  = 5.5  as  against  the  observed  value  5.7  given  in  the  table. 

For  closely  related  substances  the  agreement  of  the  #s-values 
with  one  another  is  sometimes  a very  exact  one,  as  shown  by  the 
figures  given  in  Table  VII,  which  are  based  upon  relative  surface 
tensions  determined  by  the  drop-weight  method. 


Table  VII 


Values  of  Ks=  7i  (jy'j  — 72  (/y)  3 by  the  drop- weight  method  for  several 


U~ti 


organic  liquids.  Ks  for  benzene,  CgH6,  is  taken  as  2.120. 

From  measurements  by  Morgan  and  Higgins  [Jour.  Amer.  Chem.  Soc., 


30,  1065  (1908)]. 


Substance 

■•I  CsHe 

1 C6H5C1|  CCI4 

C5HsN  1 

c6h5nh2| 

c9h7n 

k7= 

| 2.120 

| 2.120  I 2.118 

1 2.11s  1 

2.120  | 

2.125 

5.  Molecular  Weights  of  Liquids. — All  of  the  #s-values  given 
in  Tables  V,  VI,  and  VII,  were  calculated  on  the  assumption 
that  the  molecular  weight  in  the  liquid  is  the  same  as  that  in  the 
gaseous  state  in  the  case  of  every  substance  and  the  agreement 
between  the  #s-values  obtained  with  different  liquids  shows  that 
for  a great  variety  of  substances  and  especially  for  related  sub- 
stances, equation  (9)  holds  with  considerable  exactness.  It 
should  be  noted,  however,  that  just  as  good  agreement  would  be 
obtained,  if  in  each  instance  the  molecular  weight  of  the  liquid 
0 Paul  Walden,  (1863-  ).  Professor  of  Inorganic  and  PhysicalChem- 

istry  at  the  Polytechnic  Institute,  in  Riga,  Russia. 


50 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  Ill 


were  assumed  to  be  two  or  three  or  any  other  multiple  of  its 
molecular  weight  in  the  gaseous  state.  In  other  words,  the  agree- 
ment of  the  Ks-v alues  obtained  with  different  liquids  does  not 
necessarily  show  that  they  have  the  same  molecular  weight  in 
the  liquid  and  gaseous  states.  It  does,  however,  indicate  that 
if  the  molecular  weight  in  the  liquid  state  is  not  the  same  as  in  the 
gaseous  state,  then  it  must  be  some  multiple  of  the  latter  which 
multiple  is  independent  both  of  the  nature  of  the  liquid  and  of  its 
temperature.  It  would  be  rather  remarkable  if  such  a condition 
were  true  for  any  other  multiple  than  unity.  Moreover  we  have 
additional  evidence  of  quite  another  character  that  this  multiple 
is,  in  fact,  unity  and  that  all  substances  which  obey  the  Eotvos 
equation  have  the  same  molecular  formulas  in  both  the  liquid 
and  the  gaseous  states  of  aggregation.  The  Eotvos  equation 
can,  therefore,  be  employed  for  calculating  the  approximate  value 
of  the  molecular  weight  of  a pure  substance  in  the  liquid  state 
from  measurements  of  its  surface  tension  at  two  temperatures. 
The  calculation  is  most  conveniently  made  from  the  relation, 


and  the  temperature  difference,  t2  — th  should  be  fairly  large. 

Returning  now  to  section  5 of  Table  VI,  it  is  evident  that  the 
liquids  included  in  this  section  have  I£s-values  which  are  much 
smaller  than  those  of  the  other  liquids.  Moreover,  the  Ka- 
values  of  these  liquids  are  not  constant  with  respect  to  varia- 
tions in  the  temperature,  that  is,  these  liquids  do  not  obey  the 
equation  of  Eotvos.  This  is  clearly  shown  by  the  Ka  -values  for 
water  given  in  Table  VIII.  It  is  evident,  therefore,  that  since 

Table  VIII 

Showing  the  variation  of  the  Eotvos  constant,  Ks,  with  the  temperature  in 
the  case  of  water.  Calculated  on  the  basis  of  18  for  the  molecular  weight  of 
water. 

- (t)  f] =o-685  ^=o-685  £t° 


Observer 

Morgan  and  McAfee 

Ramsay  and 
Shields 

1° 

5 

15 

25 

35 

45 

55 

65 

75 

140 

K,  = 

1.08 

1.11 

1.14 

1.16 

1.20 

1.23 

1.26 

1.30 

co.  1.6 

Sec.  6] 


THE  LIQUID  STATE  OF  AGGREGATION 


51 


the  Eotvos  equation  is  not  obeyed  by  these  liquids  it  cannot  be 
logically  employed  in  calculating  their  molecular  weights.  If 
it  is  so  employed,  the  molecular  weights  thus  calculated  are  found 
to  vary  with  the  temperature  and  to  be  from  two  to  three  times 
as  large  as  the  molecular  weights  in  the  gaseous  state.  The  most 
probable  explanation  of  the  behavior  of  these  liquids  is  that  they 
are  not  pure  substances  in  the  sense  in  which  we  have  defined 
the  term  (I,  2),  that  is,  they  do  not  consist  of  a single  species  of 
molecule  but  of  a mixture  of  two  or  more  species  in  equilibrium 
with  one  another.  Indeed  there  is  abundant  evidence  from  other 
sources  that  liquid  water  is  a mixture  of  the  following  species  of 
molecules,  H20,  (H20)2,  (H20)3  and  perhaps  higher  polymers, 
all  in  chemical  equilibrium  with  one  another.  Equilibrium 
mixtures,  of  this  character  which  in  many  respects  behave  like 
pure  substances  are  usually  called  associated  substances,  to 
distinguish  them  from  pure  substances  in  the  true  sense  of  the 
term  which  are  non-associated  and  contain  but  one  species  of 
molecule.  Substances  of  the  latter  class  in  the  liquid  state  are 
also  sometimes  called  “normal”  liquids  for  the  same  reason. 

Problem  2. — From  a certain  capillary  tip  the  drop-weight  of  benzene  at 
11.4°  is  35 . 239  milligrams  and  at  68 . 5°  it  is  26.530  milligrams.  Its  densities 
at  the  same  temperatures  are  0.888  and  0.827  grams  per  cubic  centimeter 
respectively.  Calculate  the  critical  temperature  of  benzene. 

Problem  3. — From  the  same  tip  as  in  problem  (2)  the  drop-weight  of  chlor- 
benzene is  41.082  milligrams  at  8.2°  {D  = 1.120)  and  32.054  milligrams  at 
72.2°  (D  = 1.0498).  Calculate  its  molecular  weight  in  the  liquid  state. 
Calculate  also  its  critical  temperature  [tc  observed  = 360°]. 

Problem  4. — From  the  same  tip  the  drop-weight  of  aniline,  C6H5NH2,  at 
51.7°  is  45.903  milligrams  (D  = 0.9944).  Calculate  the  critical  temperature 
of  aniline  [tc  observed  =426°]. 

6.  Viscosity  and  Fluidity. — The  internal  friction  or  the  resist- 
ance experienced  by  the  molecules  in  moving  around  in  the  in- 
terior of  a body  is  termed  its  viscosity.  The  viscosity  of  gases 
is  very  low,  that  of  solids  very  high,  while  that  of  liquids  in- 
cludes a wide  range  of  variation.  Some  liquids,  like  ether,  are 
very  mobile,  while  others  such  as  pitch  and  tar  are  very  viscous. 
The  unit  of  viscosity,  called  the  coefficient  of  viscosity  or  simply 
the  viscosity,  is  represented  by  the  Greek  letter  eta,  rj,  and  is 
defined  as  the  force  required  to  move  a layer  of  the  substance  of 
unit  area,  through  a distance  of  unit  length,  in  unit  time,  past 


52 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  Ill 


an  adjacent  stationary  layer  a unit  distance  away.  The  viscosity 
of  fluids  is  commonly  measured  by  determining  the  time  of  flow 
of  a measured  volume  of  the  fluid  through  a standardized  capil- 
lary tube,  under  a definite  difference  of  pressure.  The  more 
viscous  a liquid,  the  more  slowly  it  flows. 

The  reciprocal  of  viscosity  is  called  fluidity  , 0,^0  = and  is 

sometimes  a more  convenient  quantity  to  employ  than  the  vis- 
cosity itself.  The  fluidity  is  evidently  a measure  of  the  ten- 
dency of  substances  to  flow  while  its  reciprocal,  the  viscosity,  is 
a measure  of  the  resistance  to  flow.  The  property  of  flowing 
with  considerable  ease  is  possessed  by  all  gases  and  by  most 
liquids  and  they  are  commonly  classed  as  fluids  for  this  reason. 


CHAPTER  IV 


LIQUID-GAS  SYSTEMS 

1.  The  Molecular  Kinetics  of  Vaporization. — The  molecules 
of  a liquid  are  in  a state  of  constant  unordered  motion  like  those 
of  a gas  but  they  collide  with  one  another  much  more  frequently 
owing  to  the  greater  number  of  them  in  a given  volume.  These 
collisions  take  place  without  loss  of  energy  and  although  the 
velocities  of  the  different  molecules  vary  all  the  way  from  zero 
to  very  large  values  there  is  a certain  mean  velocity  correspond- 
ing to  the  mean  kinetic  energy  ( \mu 2)  which  for  a given  liquid 
depends  only  upon  the  temperature.  This  mean  velocity  is  too 
small  to  allow  the  molecules  possessing  it  to  escape  from  a free 
surface  of  the  liquid.  There  are  always  present,  however,  some 
molecules  possessing  a much  higher  velocity  than  this  mean  ve- 
locity, so  that  if  the  liquid  is  brought  into  contact  with  a vacuous 
space,  some  of  these  rapidly  moving  molecules  will  escape  through 
the  surface  of  the  liquid  into  the  space  above  it.  In  other  words, 
some  of  the  liquid  will  evaporate.  This  evaporation  will  con- 
tinue until  the  vapor  above  the  liquid  reaches  a certain  definite 
pressure,  determined  only  by  the  temperature  and  by  the  nature 
of  the  liquid.  This  pressure  is  the  vapor  pressure  or  vapor  ten- 
sion of  the  liquid.  It  is  evidently  that  pressure  at  which  the 
rate  of  escape  into  the  gas  phase  of  the  more  rapidly  moving  liquid 
molecules  is  exactly  balanced  by  the  return  into  the  liquid  phase 
of  the  more  slowly  moving  gaseous  molecules.  Thus  while  a 
vapor  which  is  not  in  contact  with  its  liquid  may  have  any  pres- 
sure from  zero  up  to  a value  somewhat  exceeding  the  vapor  pres- 
sure, a vapor  in  contact  with  its  liquid  can  have  only  one  pressure 
at  a given  temperature,  namely,  the  vapor  pressure  of  the  liquid 
at  the  temperature. 

If  a liquid  is  brought  in  contact  with  a space  containing  some 
other  gas  at  a moderate  pressure  (say  not  exceeding  one  atmos- 
phere), then  if  this  other  gas  is  nearly  insoluble  in  the  liquid, 

53 


54 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IV 


it  follows  from  Dalton’s  law  of  partial  pressures  (II,  6)  that 
approximately  the  same  amount  of  liquid  will  evaporate  into 
the  space  as  would  if  the  second  gas  were  not  present.  Thus 
the  vapor  tension  of  water  in  contact  with  air  is  approximately 
the  same  as  its  vapor  pressure  in  a space  containing  no  second 
gas.  If  the  pressure  of  the  second  gas  is  large  or  if  it  is  appre- 
ciably soluble  in  the  liquid,  this  principle  is  no  longer  valid.  In 
other  words,  it  is  strictly  valid  only  for  the  limiting  case  of  an 
insoluble  gas  at  a very  small  pressure  and  in  any  actual  case 
it  holds  more  exactly  the  more  nearly  these  two  conditions  are 
fulfilled. 

In  Table  IX  the  vapor  pressures  of  a number  of  liquids  at  20° 
are  given. # 


Table  IX. — Vapor  Pressures  op  Various  Liquids  at  Room 
Temperature  (20°) 


Liquid 

1 

Formula 

Vapor 
pressure, 
|mm.  Hg 

Liquid 

1 1 

Formula 

Vapor 
pressure 
mm.  Hg 

Mercury 

Hg 

0.0016 

Ethyl  chloride 

C2HbC1 

996.2 

Di-amyl ! 

C10H22 

2.7 

Sulphur  dioxide  . . . 

SO2 

2,500 

Water 

H2O 

17.54 

Chlorine 

Cl  2 

5,040 

Ethyl  alcohol .... 

C2H5OH 

44.0 

Ammonia 

NH3 

6,800 

Benzene - 

CeHe 

74.7 

Hydrogen  sulphide  . 

H2S 

14,100 

Methyl  acetate ... 
Ethyl  ether 

CH3COOCH3 

(C2H5)20 

169.8 

442.0 

Carbon  dioxide 

CO2 

42,500 

2.  Heat  of  Vaporization. — Since  only  those  molecules  can 
escape  from  the  liquid  phase  which  possess  kinetic  energies  con- 
siderably greater  than  the  mean  kinetic  energy,  it  is  clear  that 
the  mean  kinetic  energy  of  the  remaining  molecules  must  have 
a lower  value  after  the  escape  of  the  more  rapidly  moving  ones. 
In  other  words  the  temperature  of  the  liquid  tends  to  fall  during 
evaporation  and  in  order  to  maintain  it  at  a constant  value, 
heat  must  be  added  to  the  liquid  from  some  external  source. 
This  heat  is  called  the  latent  heat  of  vaporization.  The  mo- 
lal  heat  of  vaporization,  Lv,  of  a liquid  under  constant  pres- 
sure, P,  at  any  temperature,  T,  is  the  quantity  of  heat  which  it 
is  necessary  to  add  to  the  liquid  in  order  to  maintain  its  tempera- 
ture at  T while  one  gram  molecular  weight  of  the  liquid  evapo- 
rates. The  heat  of  vaporization  varies  with  the  nature  of  the 
liquid,  with  its  temperature,  and  (to  a very  slight  extent)  with 
the  pressure  upon  it. 


Sec.  4] 


LIQUID-GAS  SYSTEMS 


55 


3.  Boiling  Point. — When  a liquid  is  heated  under  a definite 
external  pressure  (water  under  the  pressure  of  the  atmosphere, 
for  example),  its  vapor  pressure  increases  with  rise  in  tempera- 
ture until  finally  a temperature  is  reached  at  which  bubbles  of 
vapor  form  and  rise  from  within  the  body  of  the  liquid  and  es- 
cape into  the  space  above  it.  The  liquid  is  then  said  to  boil. 
The  boiling  point  is  defined  as  the  temperature  at  which  a liquid 
is  in  equilibrium  with  its  vapor,  when  the  vapor  pressure  is 
equal  to  the  total  pressure  upon  the  liquid.  The  temperature 
indicated  by  a thermometer  placed  in  a boiling  liquid  is  usually 
not  its  boiling  point,  for  unless  special  precautions  are  taken  a 
liquid  does  not  boil  under  equilibrium  conditions.  In  order  to 
secure  equilibrium  and  to  measure  the  temperature  correspond- 
ing to  it,  it  is  necessary  to  bring  an  intimate  mixture  of  vapor 
and  liquid  both  under  the  same  total  pressure,  into  contact  with 
the  thermometer  under  certain  conditions,  the  details  of  which 
belong  to  the  subject  of  methods  of  physical  measurements, 
which  it  is  not  the  purpose  of  this  book  to  deal  with. 

Since  the  vapor  pressure  of  a liquid  always  increases  with  in- 
crease of  temperature,  it  follows  from  the  definition  of  boiling 
point  that  it  also  must  increase  with  the  external  pressure  upon 
the  liquid.  The  boiling  point  of  a liquid  under  a pressure  of 
one  atmosphere  (76  cm.  of  Hg)  is  called  the  normal  boiling 
point  and,  unless  otherwise  specified,  it  is  this  temperature  which 
is  meant  when  the  term  boiling  point  is  used. 

4.  The  Rule  of  Ramsay  and  Young. — If  TA  and  TB  are  the 
absolute  boiling  points  of  two  related,  pure  (non-associated) 
substances  at  the  pressure,  p;  and  T\  and  Tr B their  boiling  points 
at  some  other  pressure,  p',  Ramsay  and  Young®  found  that  these 
temperatures  were  connected  by  the  following  relation,  applicable 
from  the  lowest  pressures  up  to  the  critical  point, 


(1) 


where  c is  a constant  which  is  smaller  the  more  closely  related 
the  substances  are,  being  practically  zero  for  very  closely  re- 
lated substances,  such  as  brombenzene,  C6H5Br,  and  chlor- 

a Sydney  Young  (1857-  ).  Professor  of  Chemistry  in  the  University 

of  Dublin. 


56  PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IV 

benzene,  CeHgCl,  for  example.  Table  X shows  the  values  of 
Tb 

for  a series  of  esters.  The  above  rule  evidently  requires 

that  this  ratio  should  be  constant,  if  c = 0.  It  evidently  varies 
but  slightly  for  these  closely  related  substances. 


Table  X 


Illustrating  the  Rule  of  Ramsay  and  Young  regarding  the  boiling  points 
of  closely  related  substances 


Substance 

T B 

760  mm. 

T'b 

200  mm. 

T B 

T'b 

Methylformate 

305.3° 

273 . 7° 

1.115 

Methylacetate 

330.5 

296.5 

1.115 

Methylpropionate 

352.9 

316.7 

1.114 

M ethy  lbutyrate 

375 . 3 

336.9 

1.114 

Methylvalerate 

389.7 

350.2 

1.113 

Ethylformate  

327.4 

293.1 

1.117 

Ethylacetate  

350.1 

314.4 

1.114 

Ethylpropionate  

371.3 

333.7 

1.113 

Ethylbutyrate 

392.9  • 

352.2 

1.116 

Ethylvalerate  

407.3 

365.3 

1.115 

Propylformate 

354.0 

318.0 

1.113 

Propylacetate 

373.8 

336.1 

1.112 

Propylpropionate 

395.2 

355.0 

1.113 

Propylbutyrate 

415.7 

374.2 

1.111 

Propy  lvalerate 

428.9 

385.6 

1.112 

5.  Correction  of  Boiling  Points  to  Normal  Pressure.  The 
Rule  of  Crafts. — In  practice  the  boiling  point  of  a liquid  is  usu- 
ally determined  at  the  atmospheric  pressure  prevailing  at  the 
time  the  measurement  is  made.  For  purposes  of  record  and  of 
comparison,  it  is  desirable  to  know  the  normal  boiling  point. 
The  necessary  correction  which  must  be  applied  in  order  to  re- 
duce the  observed  boiling  point  to  the  normal  pressure  can  be 
conveniently  calculated  by  means  of  a rule,  proposed  by  Crafts® 
which  is  based  upon  that  of  Ramsay  and  Young.  The  correc- 
tion, At,  to  be  added  to  a boiling  point,  TB,  on  the  absolute  scale, 
determined  at  a pressure  of  P mm,  in  order  to  reduce  it  to  760 
mm  is  given  by  the  relation, 

A t = cTBo  (760 -P)  (2) 

a James  Mason  Crafts  (1839-  ).  Professor  (retired  1900)  of  Organic 

Chemistry  at  the  Massachusetts  Institute  of  Technology. 


Sec.  5] 


LIQUID-GAS  SYSTEMS 


57 


where  the  constant,  c=™ 

l Bo 


d Tb 
d P ; 


TBo  being  the  normal  boiling 


point  of  the  liquid.  The  value  of  c for  pure  (non- associated) 
liquids  is  about  0.00012  but  it  varies  somewhat  with  the  nature 
of  the  liquid  and  in  practice  the  value  for  some  liquid  which  is 
related  as  closely  as  possible  to  the  one  under  examination  should 
be  employed  in  calculating  the  correction.  Table  XI  gives  the 
values  of  c X 104  for  a variety  of  different  liquids.  The  con- 
stant c varies  somewhat  with  P and  the  rule  of  Crafts  is,  therefore, 
accurate  only  when  760  — P is  small. 


Table  XI 


Values  of  the  constant,  c ( — ^pd  in  the  Crafts’  equation,  At  = 

cTbo  (760—  P),  for  the  correction  of  boiling  points  to  normal  pressure. 


Substance 

cXlO4 

Substance 

cXlO4 

Mercury 

1.20 

Caprylic  acid 

1.01 

Sulphur 

1.27 

Phenol 

1.0 

Tin 

1.3 

Ethyl  ether 

1 ..29 

Oxygen 

1.43 

Acetone 

1.19 

Argon 

1.35 

Propyl  acetate 

1.14 

Chlorine 

1.35 

Methyl  formate 

1.19 

Hydrochloric  acid 

1.3 

n-Hexane 

1.23 

Ammonia 

1.35 

n-Octane 

1.20 

Carbon  dioxide 

0.8 

Benzene 

1.22 

Carbon  monoxide 

1.6 

p-Xylene 

1.36 

Sulphur  dioxide 

1.0 

Diphenyl  methane 

1.25 

Carbon  disulphide 

1.1 

Dimethyl  aniline 

1.15 

Carbon  tetrachloride 

1.6 

o-Nitrotoluene 

0.98 

Stannic  chloride 

1.35 

Anthraquinone 

1.15 

Boron  trichloride 

1.3 

Benzophenone 

1.10 

Water 

1.01 

Benzoyl  chloride 

1.17 

Ethyl  alcohol 

0.97 

Camphor 

1.16 

Propyl  alcohol 

0.97 

Menthon 

1.15 

n-Butyl  alcohol 

0.90 

Naphthalene 

1.19 

Acetic  acid 

1.0 

Phthalic  anhydride 

1.19 

Proprionic  acid 

1.0 

Sulphobenzide. . . . 

1.04 

Problem  1. — The  following  boiling  points  have  been  determined  at  the 
pressures  indicated:  di-isopropyl,  60°  at  807  mm;  methyl  chloride,  —20° 
at  1.16  atmos.;  methyl  alcohol,  60°  at  625.0  mm;  germanium  tetrachloride, 
70.7°  at  0.67  atmos.;  heptylic  acid,  218.1°  at  700  mm;  camphor,  206.7° 
at  731  mm;  p-cresol,  198.5°  at  700  mm;  hydrogen  sulphide,  —60°  at  770 
mm;  zinc,  933°  at  767  mm;  bromine,  56.3°  at  700  mm;  argon,  —186.2° 
at  757.3  mm.  Calculate  the  normal  boiling  point. 


58 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IV 


6.  Trouton’s  Rule. — This  rule  states  that  for  closely  related 
substances  the  molal  heat  of  vaporization  at  the  normal  boiling 
point,  divided  by  the  normal  boiling  point  on  the  absolute  scale 
is  a constant,  that  is, 

— const . = 22  (3) 

I Bo 

A modified  Trouton’s®  rule  proposed  by  Nernst6  is  the  following, 
^ = 9.5  log  10  TBo-0.007  Tb,  (4) 

which  is  more  accurate  and  includes  a larger  variety  of  substances. 

Problem  2. — Calculate  the  heats  of  vaporization  of  six  of  the  substances 
given  in  Table  XII.  Look  up  the  experimentally  determined  values  for 
the  same  substances  and  compare.  Use  the  Landolt-Bornstein  Physikal- 
isch-Chemische  Tabellen  (4th  ed.,  1912)  and  the  “ Annual  Tables  of  Phys- 
ical and  Chemical  Constants' ' in  looking  for  the  experimental  values. 

7.  The  Critical  Phenomena. — The  significance  of  the  terms 
critical  temperature  and  critical  pressure  has  already  been 
briefly  explained  at  the  point  where  they  were  first  employed  in 
our  treatment  of  the  properties  of  gases.  We  shall  now  take  up 
in  considerable  detail  the  phenomena  which  one  observes  when  a 
pure  liquid  is  gradually  heated  in  a closed  vessel  in  contact  with 
its  vapor.  For  this  purpose  we  will  imagine  our  liquid  enclosed 
in  a transparent  cylinder  provided  with  a movable  piston.  To 
take  a concrete  case  suppose  we  start  with  liquid  isopentane,  a 
substance  which  has  been  very  fully  investigated  by  Young. 
Starting  with  1 gram  of  the  liquid  under  an  initial  pressure  of 
about  42  meters  of  mercury  (Fig.  5)  and  at  a temperature  of  160°, 
we  will  gradually  decrease  the  pressure,  keeping  the  tempera- 
ture constant,  and  in  order  to  see  clearly  the  relation  between  the 
volume  of  the  isopentane  and  the  pressure  upon  it  we  will  con- 
struct a diagram  in  which  the  pressures  will  be  plotted  as  ordinates 
and  the  corresponding  volumes  as  abscissae.  Such  a diagram  is 
shown  in  Fig.  8,  the  small  circles  representing  observed  values. 
Starting  at  the  point,  A,  therefore,  (p  = 25.2,  v = 2A)  we  gradually 

° Frederick  Thomas  Trouton,  F.  R.  S.,  Professor  of  Physics  in  the  Uni- 
versity of  London. 

6 Walther  Nernst  (1864-  ).  Professor  of  Physical  Chemistry  and 

Director  of  the  Physico-Chemical  Institute  at  the  University  of  Berlin. 


Sec.  7] 


LIQUID-GAS  SYSTEMS 


59 


decrease  the  pressure  which  follows  the  line,  AB.  When  the 
point,  B,  is  reached  where  the  pressure  is  16.3  meters  of  mercury 
the  vapor  phase  appears,  since  16.3  is  the  vapor  pressure  of  iso- 
pentane at  160°.  The  abscissa  of  this  point,  2.38  cc,  is  evidently 


Fig.  8. — Isotherms  of  Isopentane. 

the  specific  volume  of  liquid  isopentane  at  160°  when  under  its 
own  vapor  pressure.  It  is  called  the  orthobaric  specific  volume 
of  the  liquid.  As  we  continue  to  raise  the  piston  the  liquid  con- 
tinues to  evaporate  (Fig.  4)  and  the  total  volume  consequently  in- 

5 


60 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IV 


creases  along  the  line,  BC,  while  the  pressure  remains  perfectly 
constant.  When  the  point,  C,  is  reached  all  of  the  liquid  has 
evaporated  and  the  abscissa  of  this  point,  13.72  cc,  is  therefore 
the  specific  volume  of  the  saturated  vapor  at  this  temperature. 
It  is  called  the  orthobaric  specific  volume  of  the  vapor.  If  we 
now  continue  to  raise  the  piston  (Fig.  3)  the  pressure  falls  and  the 
volume  increases  as  shown  by  the  curve  CD,  and  this  curve 
would  continue  in  this  direction  until  the  pressure  falls  to  zero 
and  the  volume  becomes  infinite.  Along  the  portion,  AB,  there- 
fore, we  have  only  liquid  isopentane  and,  as  is  characteristic  of 
the  liquid  state,  the  volume  is  seen  to  change  but  slightly  with 
change  in  pressure.  Along  the  portion,  BC,  we  have  both  liquid 
and  vapor  in  equilibrium  at  the  constant  pressure,  16.3,  which  is 
the  vapor  pressure  of  the  liquid  at  this  temperature.  Finally 
along  the  portion,  CD,  we  have  only  the  vapor,  and  its  volume 
decreases  approximately  in  inverse  proportion  to  the  pressure,  as 
required  by  the  perfect  gas  law.  The  whole  curve,  ABCD,  is 
called  the  isothermal  for  160°. 

If  we  now  raise  the  temperature  to  170°  and  repeat  the  above 
process,  we  obtain  the  isothermal  labeled  170°,  which  differs 
from  the  previous  one  chiefly  in  the  respect  that  the  horizontal 
portion,  BC,  is  shorter,  that  is,  the  orthobaric  specific  volumes 
and,  therefore,  the  orthobaric  densities  of  liquid  and  vapor  are 
much  nearer  together  than  at  160°.  Passing  then  to  the  iso- 
thermal for  180°  we  see  the  two  orthobaric  specific  volumes  ap- 
proaching each  other  still  closer,  that  of  the  liquid  increasing  and 
that  of  the  gas  decreasing,  until  finally  as  we  continue  to  raise 
the  temperature  we  find  that  on  the  isothermal  for  187.8°,  the 
two  volumes  coincide  and  the  portion,  CD,  of  the  curve  is  re- 
duced to  a point.  At  this  point,  called  the  critical  point  of  the 
substance,  all  the  physical  properties  of  the  two  phases  become 
identical  and  the  distinction  between  gas  and  liquid  disappears. 
This  fact  is  made  evident  to  the  observer  by  the  gradual  flatten- 
ing and  final  disappearance  of  the  meniscus  which  separates  the 
two  phases,  showing  that  with  rising  temperature  the  surface 
tension  of  the  liquid  gradually  decreases  and  finally  becomes 
zero  at  the  critical  point.  The  temperature  of  the  critical  iso- 
therm is  called  the  critical  temperature,  tc,  of  the  substance. 
The  corresponding  pressure,  which  is  evidently  the  vapor  pres- 


Sec.  7] 


LIQUID-GAS  SYSTEMS 


61 


sure  of  the  liquid  at  the  critical  temperature  as  well  as  the  maxi- 
mum vapor  pressure  which  the  liquid  can  have,  since  it  ceases 
to  exist  at  this  point,  is  called  the  critical  pressure,  pc,  of  the 
substance.  The  density  of  the  substance  at  the  critical  point 
is  called  the  critical  density,  Dc,  and  the  reciprocal  of  the  density 
or  the  specific  volume  is  called  the  critical  volume,  vc.  The 
' critical  constants  of  a number  of  substances  are  given  in  Table 
XII.  Above  its  critical  temperature  no  gas  can  be  caused  to 
liquefy  no  matter  how  great  a pressure  is  put  upon  it.  In  Fig. 
8 the  isothermals  above  the  critical  point  show  at  first  a flatten- 
ing in  the  neighborhood  of  the  critical  volume.  With  increasing 
temperature,  however,  this  flattening  gradually  disappears  and 
the  isothermals  smooth  out  into  the  hyperbolas  of  a perfect  gas, 
hyperbolas  which,  in  other  words,  are  graphs  of  the  equation, 

pv  = const. 


Table  XII. — Physical  Properties  of  some  of  the  Principal  Gases 

and  Vapors 


Gas 

Density 

under 

standard 

conditions 

D 

g.  per  liter 

Melting 

point, 

tf 

Boiling 

point, 

TBv 

760  mm. 

Critical 

tempera- 

ture, 

Tc 

Critical 

pressure, 

Vc 

atmos- 

pheres 

Critical 

density, 

Dc 

grams 

per  cc 

He 

0.1785 

V 

4.25° 

5.25° 

2.26 

0.07 

Hz 

0.08987 

14.1 

20.3 

312 

13. 42 

0.033 

Ne 

0.9002 

30 

53 

55 

29. 

N2 

1 . 2507 

62.5 

77.3 

127 

35.0 

0.327 

A 

1 . 7809 

85.1 

87.2 

150.7 

48.0 

0.509 

02 

1.4290 

46 

90.1 

154.2 

50.8 

0.4292 

NO 

1 . 3402 

112.5 

122.5 

179.5 

71.2 

CH4 

0.7168 

89 

108.3 

191.2 

54.9 

Kr 

3.708 

104 

121.3 

210.5 

54.3 

Xe 

5.851 

133 

163 

289.0 

58.0 

1.15 

C02 

1.9768 

216.3 

194.7 

304.1 

72.9 

0.448 

N20 

1.9777 

170.3 

183.2 

309.5 

71.65 

HC1 

1 . 6394 

161.6 

189.9 

324.5 

81.6 

0.41 

NHs 

0.7708 

194.8 

239.5 

406.0 

112.3 

CI2 

3. 16741 

171.0 

239.4 

417.0 

79.6 

SO2 

2.9266 

200.7 

263 

430.3 

77.7 

0.513 

n-CsHi2 

liquid 

125.6 

309.3 

470.3 

33.026 

0.2323 

n-C7H,6 

liquid 

176.0 

371.5 

540.0 

26.881 

0.2341 

CeHsF 

liquid 

..2.ZZ.-9 

358.3 

559.7 

44.619 

0.3541 

CeHe 

liquid 

278.5 

353.4 

561.7 

47.89 

0.3045 

SnCl4 

liquid 

240 

387.2 

591.8 

36.95 

0.7419 

CeHsCI 

liquid 

228 

405.1 

632.3 

44.631 

0.3654 

H2O 

liquid 

273.1 

373.1 

647.0 

217.5 

0.4 

Hg 

liquid 

235.2 

630.1 

>1550.0 

1 Jaquerod  and  Tourpaian  [J.  Chim.  Phys.,  11,  274  (1913)]  give  3.214. 

2 Bulle  [Physik.  Z.,  14,  860  (1913)]  gives  32°  and  11.0  atmos. 


62 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IV 


Problem  3. — What  is  the  value  of  the  latent  heat  of  vaporization  of  a 
liquid  just  below  its  critical  temperature? 

Problem  4. — Describe  a process  by  which  it  is  possible  to  start  with  liquid 
water  at  10°  under  a pressure  of  one  atmosphere  and  convert  it  completely 
into  steam  at  110°,  without  causing  it  to  “evaporate,”  that  is,  without  at 
any  time  having  two  phases  in  the  system. 


8.  The  Rule  of  Cailletet d and  Mathias.6 — This  rule  states  that 
the  mean  of  the  two  orthobaric  densities  of  a pure  (non-asso- 
ciated)  substance  is  a linear  function  of  the  temperature,  or 
mathematically, 


Dl~\~Dg 

2 


= a+bT 


(5) 


This  is  illustrated  graphically  in  Fig.  9 which  is  self  explanatory. 


Fig.  9. — Illustrating  the  Rule  of  Cailletet  and  Mathias.  The  points  along 
CD  represent  the  means  of  the  two  corresponding  orthobaric  densities. 

One  of  the  principal  uses  of  this  rule  is  in  determining  the  value 
of  the  critical  density  of  a substance,  as  it  is  usually  not  possible 
to  measure  this  density  directly.  The  method  of  applying  the 
rule  to  such  cases  is  illustrated  by  problem  5.  As  a rule  the  re- 
lation holds  more  exactly  in  the  immediate  neighborhood  of  the 
critical  point  than  it  does  at  lower  temperatures. 
a Louis  Cailletet,  Iron  Master  at  Chatillon-sur-Seine. 

6 E.  Mathias,  Professor  of  Physics  and  Meteorology  at  the  University  of 
Clermont,  France. 


Sec.  9] 


LIQUID-GAS  SYSTEMS 


Problem  5. — The  critical  temperature  of  normal  pentane  is  197.2°.  At 
150°  the  orthobaric  densities  of  this  substance  are  0.4604  and  0.0476 
respectively.  At  190°  they  are  0.3445  and  0.1269.  Calculate  the  critical 
density  of  normal  pentane.  [Observed  value,  0.2323.] 

9.  Superheating  and  Supercooling. — When  a body  of  liquid  is 
gradually  heated  from  the  outside  it  is  frequently  possible  to  raise 
its  temperature  considerably  above^v^&iling  point — as  much  as 
200°  in  the  case  of  water.  Such^t&quid  is  said  to  be  superheated. 
When  a liquid  is  heated  in  a vessel  by  the  application  of  heat  to 
the  bottom  of  the  vessetyihe  lower  layers  of  the  liquid  become 
m time  to  time  portions  of  these  superheated 


vapor  with  explosive  violence  causing  the 


phenomenon  known  as  “bumping.”  Superheating  and  hence 
bumping  can  be  prevented  or  greatly  reduced  by  having  some  of 
the  vapor  phase  in  contact  with  the  liquid  at  the  point  where 
heat  is  applied.  In  practice  this  is  usually  attained  by  intro- 
ducing pieces  of  capillary  tubing  or  of  porous  porcelain  plate. 
The  small  capillaries  become  filled  with  vapor  and  the  presence 
of  the  vapor  tends  to  prevent  superheating  of  the  liquid  which 
thus  boils  quietly,  the  bubbles  of  vapor  forming  and  rising  from 
the  point  of  contact  of  the  liquid  with  the  vapor  held  in  the 
capillaries. 

The  reverse  of  superheating,  that  is  supercooling,  is  fre- 
quently observed  when  a gas  or  vapor  is  cooled.  It  may  be 
cooled  considerably  below  the  temperature  at  which  its  pressure 
becomes  equal  to  the  vapor  pressure  of  the  liquid,  without  the 
appearance  of  any  liquid  phase  in  the  system.  If  a single  drop  of 
liquid  be  introduced,  however,  condensation  immediately  occurs. 
In  fact  as  a general  rule  when  any  phase,  whether  liquid,  gas,  or 
solid,  reaches  a condition  where  it  ought  normally  to  change  over 
into  another  state  of  aggregation  or  another  phase,  this  change 
frequently  does  not  take  place  at  once  but  supercooling  or  super- 
heating occurs  instead.  The  introduction  of  a trace  of  the 
second  phase,  however,  is  usually  sufficient  to  bring  about  the 
change  and  to  prevent  any  great  degree  of  supercooling  or  super- 
heating. Mechanical  agitation  or  the  presence  of  small  particles, 
such  as  dust,  frequently  has  a similar  effect. 


64 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IV 


REFERENCES 

Books:  (1)  Young’s  Stoichiometry,  Chapters,  VIII,  IX  and  X. 

Journal  Articles:  (2)  Theodor  Lohnstein,  Z.  physik.  Chem.  84,  410 
(1913).  (3)  Morgan,  Jour.  Amer.  Chem.  Soc.,  1908  to  1914.  (4)  Walden 

and  Swinne,  Z.  Physik.  Chem.  82,  290  (1913). 


CHAPTER  V 


THE  CRYSTALLINE  STATE  OF  AGGREGATION 

1.  General  Characteristics  of  the  Crystalline  State. — When  a 
substance  in  the  gaseous  state  at  any  temperature  below  its  melt- 
ing point  is  subjected  to  gradually  increasing  pressures,  or  when 
a substance  in  the  liquid  state  is  gradually  cooled,  a point  will 
usually  be  reached  where  a third  state  of  aggregation,  the  crys- 
talline state,  appears  in  the  system.  This  state  of  aggregation 
is  characterized  by  a very  slight  compressibility,  usually  much 
smaller  even  than  that  of  the  liquid  state.  Examination  of  any 
substance  in  the  crystalline  state  shows  that  it  is  made  up  of  an 
aggregation  of  individuals  having  a definite  geometric  form,  which 
form  is  one  of  the  characteristic  properties  of  the  substance. 
These  geometric  forms  are  called  crystals  and  the  physical  prop- 
erties of  these  crystals  bear  a close  connection  to  the  crystalline 
form.  A crystal  may,  in  fact,  be  defined  as  a homogeneous  body 
possessing  definite  and  characteristic  vector  properties,  that  is, 
properties  which  are  different  in  different  directions  through  the 
crystal.  Such  a body  is  called  an  anisotropic  body  (I,  8).  Thus, 
for  example,  the  conductivity  for  heat  or  for  electricity  measured 
in  one  direction  through  a crystal  may  be  different  from  that 
measured  in  a direction  at  right  angles  to  the  first,  and  the  index 
of  refraction  for  light  usually  depends  upon  the  direction  in  which 
the  light  is  sent  through  the  crystal.  The  systematic  descrip- 
tion and  classification  of  the  different  geometric  forms  displayed 
by  crystals  and  the  relation  of  these  forms  to  the  physical  prop- 
erties of  the  crystals  belongs  to  the  subject  of  crystallography, 
which  will  not  be  entered  upon  in  this  book. 

In  the  case  of  most  crystalline  substances  the  crystals  possess 
rigidity,  that  is,  they  offer  a resistance  to  deformation  by  mechan- 
ical force.  For  this  reason  they  are  called  solid  substances  and 
because  the  property  of  solidity  is  common  to  the  large  majority 
of  crystalline  substances,  the  crystalline  state  is  commonly 

65 


66 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  V 


spoken  of  as  the  solid  state.  The  property  of  solidity  is  by  no 
means  restricted  to  crystals,  however,  but  is  possessed  to  a very 
pronounced  degree  by  some  liquids.  Moreover,  there  are  some 
crystals  which  possess  scarcely  any  appreciable  solidity.  The 
term  crystalline  state  is,  therefore,  a better  name  for  this  state 
of  aggregation  than  the  more  customary  one  of  “ solid  state.” 

When  a crystalline  substance  is  gradually  heated  under 
constant  pressure  (that  of  the  atmosphere,  for  example)  it  either 
completely  evaporates,  thus  changing  into  a gas  or  vapor,  or 
else,  if  the  pressure  upon  it  be  sufficiently  great,  it  loses  its 
crystalline  form  and  changes  into  the  liquid  state,  as  soon  as  a 
definite  temperature  is  reached.  This  temperature,  which  de- 
pends to  a slight  extent  upon  the  pressure,  is  called  the  melting 
point  of  the  substance  and  is  one  of  its  most  characteristic  prop- 
erties. The  molecular  kinetics  of  the  melting  process  and  the 
probable  molecular  condition  within  a crystal  will  be  considered 
in  the  next  chapter. 

2.  Liquid  Crystals. — Most  crystals  are  rigid  and  will  fracture 
when  subjected  to  pressure,  but  many  substances  are  known  to 
form  crystals  in  which  the  crystal  forces  are  so  weak  that  the 
crystals  can  be  easily  distorted  and  will  even  flow,  form  drops 
and  rise  in  capillary  tubes  under  the  influence  of  the  sur- 
face tension  forces.  They  are  called  liquid  crystals1, 2 or  crys- 
talline liquids.  In  common  with  solid  crystals,  however, 
they  possess  a definite  melting  point  and  the  optical  properties 
characteristic  of  the  crystalline  state.  The  property  of  flowing 
under  pressure  is  not  confined  to  substances  usually  called  liquid 
crystals,  however,  but  is  probably  possessed  to  some  extent  by 
all  crystals.  Thus  ice,  which  will  fracture  if  struck,  will  gradually 
flow,  if  subjected  to  great  pressure.  This  gradual  flow  is  prob- 
ably at  least  partially  responsible  for  the  movement  of  glaciers 
down  mountain  sides  and  through  valleys. 

3.  Polymorphism  and  Transition  Point. — When  a crystalline 
substance  is  gradually  heated  we  find  in  some  cases  that  when  a 
certain  temperature  is  reached  a change  occurs  in  which  this 
crystal  form  disappears  and  a second  crystal  form  appears.  That 
is,  the  substance  has  the  property  of  existing  in  more  than  one 
form  of  crystals.  This  property  is  known  as  polymorphism. 
The  transition  of  one  form  into  the  other  is  usually  attended  by 


Sec.  4]  TIIE  CRYSTALLINE  STATE  OF  AGGREGATION 


67 


a pronounced  volume  change  and  an  appreciable  evolution  or 
absorption  of  heat.  The  two  forms  of  crystals  are  perfectly 
distinct  and  must  be  regarded  as  different  crystalline  phases.  At 
a certain  temperature,  called  the  transition  or  inversion  tempera- 
ture, these  two  phases  can  exist  in  equilibrium  with  each  other. 
Above  this  temperature  one  form  only  is  stable  and  below  this 
temperature  the  other  form  only  is  stable.  The  quantity  of  heat 
absorbed  when  one  mole  of  a substance  in  the  crystalline  form, 

A,  changes  to  the  form,  B,  at  the  transition  temperature  is  called 
the  molal  heat  of  transition  from  the  form,  A,  to  the  form, 

B,  at  this  temperature.  The  transition  temperature  varies 
slightly  with  the  pressure. 

The  nature  of  the  molecular  kinetics  of  the  process  of  transi- 
tion may  be  inferred  from  that  of  the  process  of  fusion  which 
will  be  described  in  the  next  chapter.  Owing  to  the  restricted 
character  of  the  molecular  motion  in  a crystal  (see  VII,  1),  the 
process  of  transition  from  one  form  to  the  other  is  frequently  a 
very  slow  one  and  if  the  transition  temperature  happens  to  be 
rather  high,  both  forms  can  be  kept  for  practically  an  indefinite 
period  at  ordinary  temperatures  although  only  one  of  these  forms 
is  strictly  stable  under  these  conditions.  The  unstable  form  has 
still  the  tendency  to  change  over  to  the  other  even  though  no 
change  can  be  observed  over  long  periods  of  time.  (Cf.  Super- 
cooling of  Liquids,  IV,  9).  Thus  two  crystalline  modifications 
of  calcium  carbonate,  calcite  and  aragonite,  are  found  in  nature 
although  only  the  former  is  “ stable”  at  ordinary  temperatures. 

4.  Isomorphism  and  the  Rule  of  Mitscherlich. — It  frequently 
happens  that  two  substances  of  analogous  composition  such  as 
arsenic  acid,  H3As04,  and  phosphoric  acid,  H3P04,  form  crystals 
which  resemble  each  other  very  closely,  so  closely  in  fact  that 
when  a crystal  of  one  is  placed  in  a solution  of  the  other  the 
second  will  crystallize  out  upon  the  first  and  from  a solution 
containing  both  substances  mixed  crystals,  that  is,  homogeneous 
crystals  containing  both  substances,  can  be  obtained.  Such 
substances. are  said  to  be  isomorphous.  On  the  basis  of  this  be- 
havior Mitscherlich®  proposed  a rule  which  was  once  of  con- 

° Eilhardt  Mitscherlich  (1794-1863).  The  son  of  a preacher.  He  de- 
voted himself  first  to  the  study  of  history  philology  and  oriental  languages. 
In  1821  he  became  Professor  of  Chemistry  in  the  University  of  Berlin. 


68 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  V 


siderable  importance  in  deciding  which  multiple  of  the  combining 
weight  of  an  element  was  its  atomic  weight.  This  rule  may  be 
stated  as  follows:  Two  isomorphous  substances  have  analogous 
molecular  formulas.  Thus,  in  the  case  cited  above,  if  the  formula 
of  phosphoric  acid  were  known  to  be  H3PO4,  then  we  could  in- 
fer that  that  of  arsenic  acid  was  H3ASO4  and  not  H3AS2O4  or 
H6AsOs,  for  example,  and,  therefore,  if  the  per  cent,  of  arsenic 
in  the  compound  were  determined,  its  atomic  weight  could  be 
calculated.  The  rule  of  Mitscherlich  was  used  quite  extensively 
by  Berzelius®  for  fixing  atomic  weights  but  is  seldom  employed 
to-day  as  it  is  not  a very  safe  one,  several  very  marked  exceptions 
to  it  being  known,3  and  also  because  we  now  have  much  more 
satisfactory  methods  which  can  be  used  for  the  same  purpose. 

5.  Relations  between  Crystal  Form  and  Chemical  Constitu- 
tion.— Mitscherlich’s  rule  of  isomorphism  was  one  of  the  earliest 
attempts  to  correlate  crystal  form  with  chemical  constitution. 
In  recent  years  the  subject  has  again  received  considerable 
attention  and  a theory  has  been  formulated  by  means  of  which 
many  of  the  forms  displayed  by  crystals  and  the  resemblances 
and  differences  exhibited  by  the  crystals  of  different  substances 
can  be  interpreted  in  terms  of  the  known  properties  of  the  chem- 
ical elements  which  make  up  the  crystal.  This  theory  was  first 
advanced  by  Barlow h and  Pope c and  was  based  upon  the  assump- 
tion that  every  crystal  is  a close-packed  assemblage  of  atomic 
spheres  which  can  be  partitioned  into  small  cells,  all  exactly 
similar  and  all  marshalled  in  rows  and  columns  thus  giving  the 
symmetrical  form  to  the  crystal.  These'  small  cells  are  the 
chemical  molecules,  and  the  atoms  and  molecules  are  assumed  to 
be  capable  of  a certain  amount  of  distortion  under  the  influence 
of  the  forces  acting  between  them.  This  theory  has  been  recently 
modified  and  improved  by  T.  W.  Richards  who  introduced  the 

0 Jons  Jacob  Berzelius  (1779-1848).  The  son  of  a school  master.  Grad- 
uated in  medicine  from  the  University  of  Upsala  and  became  a practising 
physician  in  Stockholm.  In  1806  became  Professor  of  Chemistry  in  the 
University  of  Stockholm.  Discovered  cerium,  selenium  and  .thorium  and 
determined  the  atomic  weights  of  many  of  the  elements.  Introduced  the 
present  set  of  symbols  for  the  elements. 

b William  Barlow  (1845-  ).  English  Chemist. 

c William  Jackson  Pope,  F.  It.  S.  (1870-  ).  Professor  of  Chemistry 

in  Cambridge  University. 


Sec.  6]  THE  CRYSTALLINE  STATE  OF  AGGREGATION 


69 


additional  assumption  (for  which  there  exists  considerable  con- 
formatory  evidence  from  other  sources)  that  the  atoms  were 
capable  of  actual  compression  as  well  as  distortion.  For  a more 
detailed  description  of  this  theory  and  examples  of  its  application 
to  specific  cases  the  student  should  consult  the  papers  of  Richards 
and  of  Barlow  and  Pope  cited  below.  With  the  assistance  afforded 
by  a new  system3  for  the  quantitative  classification  of  crystals 
perfected  by  Federov,®  the  theories  of  Barlow  and  Pope  and  of 
Richards  should  mark  the  beginning  of  a new  epoch  in  the  subject 
of  chemical  crystallography. 

6.  The  Internal  Structure  of  Crystals. — A powerful  stimulus 
has  recently  been  given  to  the  old  question  of  the  arrangement 
of  the  atoms  and  molecules  within  the  crystal  network,  by  the 
results  obtained  from  the  study  of  the  reflection  and  refraction 
of  X-rays  by  crystals.  X-rays  are  now  recognized  to  be  of  the 
same  nature  as  ordinary  light  rays  but  to  have  extremely  short 
wave  lengths.  It  occurred  to  Laue,6  that  if  this  were  the  case 
the  successive  rows  of  molecules  in  a crystal  ought  to  behave 
toward  these  very  short  waves  in  the  same  way  as  a grating  spec- 
troscope does  to  ordinary  light  rays,  that  is,  diffraction  effects 
and  X-ray  spectra  ought  to  be  obtained.  Experiment  fully 
confirmed  this  conclusion  and  is  not  only  yielding  important 
information  regarding  the  X-rays  themselves  but  is  also  giving 
us  pictures  of  the  internal  structure  of  crystals.6  When  X-rays 
after  reflection  or  refraction  by  a crystal  are  allowed  to  impinge 
upon  a photographic  plate  or  a fluorescent  screen,  patterns  are 
produced  whose  form  changes  as  the  crystal  is  rotated  in  different 
directions  and  from  a study  of  the  patterns  thus  obtained,  with 
the  crystal  held  in  different  positions,  the  arramgement  of  the  atoms 
and  molecules  within  the  crystal  can  be  determined.  In  this 
way  it  has  been  found7  by  Bragg  and  Bragg c that  in  the  diamond 
each  carbon  atom  throughout  the  whole  crystal  is  surrounded  by 

° Evgraf  Stephanovic  Federov,  Professor  of  Mineralogy  in  the  Institute 
of  Mines  of  the  Empress  Catherine  II,  at  Petrograd. 

6 Max  von  Laue,  Professor  of  Theoretical  Physics  in  the  University  of 
Zurich,  Switzerland. 

c William  Henry  Bragg,  F.  R.  S.  (1862-  ).  Professor  of  Mathematics 

and  Physics  in  Leeds  University,  England.  His  son,  W.  Lawrence  Bragg, 
Investigator  in  the  Cavendish  Laboratory,  Cambridge,  England. 


70 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  V 


four  others  so  as  to  form  a regular  tetrahedron,  a discovery 
which  is  of  much  interest  to  the  chemist.  This  method  of  in- 
vestigating crystal'  structure  is  still  in  its  infancy  but  promises 
to  yield  many  important  results. 

REFERENCES 

Books:  (1)  Die  neue  Welt  der  fliissigen  Kristalle  und  deren  Bedeutung 
fur  Physik,  Chemie,  Technik  und  Biologie,  Lehmann,  1911.  (2)  Kristall- 

inischfliissige  Substanzen.  Vorlander,  1908. 

Journal  Articles:  (3)  Federov,  Z.  Kryst.  Min.,  52,  11,  22,  and  97 
(1914).  (4)  T.  W.  Richards,  Jour.  Amer.  Chem.  Soc.,  36,  382  and  1686 

(1914).  (5)  Barlow  and  Pope,  Ibid.,  36,  1675  and  1694  (1914).  (6)  W.  L. 
Bragg,  The  Analysis  of  Crystals  by  the  X-ray  Spectrometer.  Proc.  Roy. 
Soc.  Lon.,  A,  89,  468  (1914).  (7)  Bragg,  Ibid.,  A,  89,  277  (1914). 


CHAPTER  VI 


CRYSTAL-GAS  SYSTEMS 

1.  Vapor  Pressure  of  Crystals. — If  the  pressure  upon  a crystal 
at  constant  temperature  is  gradually  reduced,  a point  will  even- 
tually be  reached  where  the  gaseous  phase  appears  in  the  system, 
that  is,  the  crystal  begins  to  evaporate  in  much  the  same  way  as 
a liquid  does  under  similar  circumstances.  The  pressure  at 


which  the  gaseous  phase  and  the  crystalline  phase  are  in  equilib- 
rium with  each  other  at  any  temperature  is  called  the  vapor 
pressure  or  sublimation  pressure  of  the  crystals  at  that  tempera- 
ture. If  the  processes  described  in  section  7 of  Chapter  IV  be 
carried  out  with  a crystalline  substance,  starting  at  a temperature 
considerably  below  its  melting  point,  the  character  of  the  isother- 

71 


72 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  VI 


mals  obtained  will  at  first  resemble  very  closely  those  of  a liquid 
as  shown  in  Fig.  8.  With  rising  temperature  the  densities  of 
the  two  phases,  crystalline  and  gas,  gradually  approach  each 
other.  Long  before  a temperature  is  reached  where  they  come 
very  near  together,  however,  the  crystalline  phase  usually  melts 
and  thus  disappears  from  the  system.  We  might  imagine  a 
case,  however,  in  which  the  melting  point  lay  at  such  a high  tem- 
perature that  before  it  was  reached  the  densities  of  the  two 
phases  became  identical.  The  isothermal  for  this  temperature 
would  then  have  the  same  general  form  as  the  one  marked  AO  A' 
in  Fig.  10.  Identity  with  respect  to  density  does  not,  however, 
necessarily  imply  that  all  the  other  physical  properties  of  the 
two  phases  are  identical.  The  temperature  at  which  the  densi- 
ties become  equal  would  not,  therefore,  necessarily  be  a critical 
temperature,  above  which  the  crystalline  state  of  aggregation 
could  not  exist.  In  fact  the  isothermals  above  this  temperature 
in  the  crystal-gas  system  would  not  resemble  those  for  the  liquid- 
gas  system,  as  shown  in  Fig.  8,  but  would  instead  probably  be 
somewhat  of  the  general  character  of  those  shown  in  Fig.  10. 

2.  Sublimation  Point. — The  temperature  at  which  the  vapor 
pressure  of  a crystal  becomes  equal  to  the  external  pressure  is 
evidently  perfectly  analogous  to  the  boiling  point  of  a liquid  and 
may  be  called  the  “sublimation  point”  of  the  crystal.  In  the 
case  of  most  crystalline  substances  the  “normal  sublimation 
point,”  or  the  sublimation  point  under  atmospheric  pressure, 
lies  above  the  melting  point  of  the  substance  and  is  consequently 
never  reached  in  practice.  With  some  substances,  such  as  arsenic 
trioxide,  however,  the  melting  point  is  the  higher  of  the  two 
temperatures  and  hence  these  substances  when  heated  simply 
sublime  or  “boil”  away  completely  without  melting.  They 
cannot  be  melted  except  at  higher  pressures  than  that  of  the 
atmosphere. 

3.  Heat  of  Sublimation. — The  process  of  vaporization  of  a 
crystal,  like  that  of  a liquid,  is  attended  by  an  absorption  of 
heat  and  the  amount  of  heat  absorbed  when  one  mole  of  the 
crystals  vaporizes  at  a given  temperature  and  pressure  is  called 
the  molal  heat  of  vaporization  or  of  sublimation,  Ls,  of  the 
crystals  at  that  temperature  and  pressure. 


CHAPTER  VII 


CRY STAL -LIQUID  SYSTEMS 

1.  The  Molecular  Kinetics  of  Crystallization  and  Fusion. — 

The  following  hypothetical  picture  of  the  mechanism  of  crys- 
tallization and  fusion  is  in  accordance  with  the  known  facts  con- 
cerning the  process  and  will  help  the  student  to  appreciate  the 
probable  difference  in  the  molecular  condition  of  the  two  states 
of  aggregation. 

We  have  already  seen  (IV,  1)  that  the  molecules  of  a liquid 
which  have  velocities  considerably  higher  than  the  velocity 
corresponding  to  the  mean  kinetic  energy  are  able  to  escape  from 
the  field  of  attraction  of  the  other  molecules  and  to  enter  the 
vapor  phase.  There  are  also  present  in  the  liquid  numbers  of 
molecules  having  velocities  considerably  smaller  than  this  mean 
velocity.  When  several  of  these  slowly  moving  molecules  come 
together  the  crystal  forces,  that  is,  the  forces  which  hold  the  mole- 
cules together  in  the  crystalline  state,  may  be  strong  enough  to 
prevent  the  molecules  from  flying  apart  again  after  the  impact. 
These  molecules  under  the  influence  of  the  crystal  forces  may 
then  arrange  themselves  into  the  form  of  a minute  crystal  which 
thereafter  moves  as  a unit  until  it  collides  with  some  rapidly 
moving  molecule  and  is  broken  up  by  the  impact  or  else  meets 
with  other  slowly  moving  molecules  which  it  attracts  and  holds, 
thus  growing  in  size.  If  heat  be  gradually  and  continuously 
abstracted  from  the  liquid,  the  temperature  falls  and  the  mean 
kinetic  energy  of  the  molecules  decreases.  This  will  evidently 
increase  the  chances  of  formation  of  these  minute  crystals,  called 
crystal  nuclei,  and  will  increase  their  average  life  and  chances  of 
growth.  Finally  a temperature  will  be  reached  at  which  the 
mean  kinetic  energy  of  the  liquid  molecules  becomes  so  small 
that  the  chances  of  formation  and  growth  of  the  crystal  nuclei 
are  just  equal  to  their  chances  of  destruction.  That  is,  the  chance 
that  any  crystal  nucleus  taken  at  random  will  continue  to  grow 

73 


74 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  VII 


by  the  addition  to  it  of  the  more  slowly  moving  molecules  with 
which  it  collides  is  just  equal  to  the  chance  that  it  will  be  broken 
up  by  collision  with  more  rapidly  moving  molecules. 

If  the  temperature  be  then  further  reduced,  it  is  evident  that 
the  chances  of  growth  will  be  greater  than  the  chances  of  de- 
struction and  if  a crystal  of  the  substance  be  now  introduced  into 
the  liquid  and  the  abstraction  of  heat  be  continued,  this  crystal 
will  be  observed  to  increase  in  size  at  the  expense  of  the  liquid. 
In  other  words  the  liquid  is  said  to  crystallize  or  to  freeze.  If 
we  do  not  introduce  a crystal  into  the  liquid,  the  crystalline  phase 
may  eventually  form  spontaneously  at  several  points  within  the 
liquid.  In  either  case  as  soon  as  the  crystalline  phase  appears, 
it  will  be  noticed  that  the  further  withdrawal  of  heat  from  the 
system  fails  to  produce  any  corresponding  decrease  in  its  tem- 
perature. It  tends  to  cause  a lowering  in  the  mean  kinetic  energy 
of  the  molecules  of  the  liquid,  it  is  true,  but  if  the  liquid  is  in  con- 
tact with  the  crystalline  phase,  the  slower  moving  molecules  of 
the  liquid  are  attracted  and  held  by  the  crystals  which  thus  grow 
in  size  at  the  expense  of  the  liquid  phase,  the  mean  kinetic  energy 
in  the  latter  phase  and  hence  also  its  temperature  remaining 
perfectly  constant  even  though  heat  be  continuously  abstracted 
from  it. 

Problem  1. — Compare  this  process  with  the  mechanism  by  which  the 
temperature  of  a liquid  boiling  under  constant  pressure  remains  perfectly 
constant  even  though  heat  be  continuously  added  to  it. 

This  constant  temperature  at  which  the  liquid  and  crystalline 
phases  are  in  equilibrium  with  each  other  is  called  the  freezing 
point  of  the  liquid  or  the  melting  point  of  the  crystals. 

Let  us  now  consider  the  reverse  process  where  we  start  with  a 
crystal  at  a low  temperature  and  gradually  add  heat  to  it.  The 
molecules  of  the  crystal  are  held  in  the  crystal  network  by  the 
action  of  the  crystal  forces.  They  are  not  at  rest,  however,  but 
oscillate  or  vibrate  about  a certain  mean  position.  The  plane 
of  oscillation  and  the  amplitude  may  vary  from  molecule  to  inole- 
cule  but  there  is  probably  a mean  amplitude  of  oscillation  which 
like  the  mean  kinetic  energy  of  liquid  or  gas  molecules  is  depend- 
ent upon  the  temperature.  The  molecular  motion  within  the 
crystal  is,  therefore,  unordcred,  being  in  all  sorts  of  directions 


Sec.  2] 


CRYSTAL-LIQUID  SYSTEMS 


75 


and  having  all  amplitude  values  on  both  sides  of  that  average 
amplitude  which  corresponds  to  the  temperature  of  the  crystal. 
The  motion  is  not  “free  random  motion”  like  that  in  a gas  or 
a liquid,  however,  but  is  instead  “restrained  random  motion” 
since  a molecule  cannot  move  freely  about  among  the  other 
molecules,  but  can  only  oscillate  about  a more  pr  less  definite 
center  in  the  crystal  network.  From  time  to  time,  however, 
i some  of  the  molecules  will  attain  amplitudes  of  oscillation  so 

great  that  they  will  get  far  enough  from  the  crystal  network 
to  escape  from  the  crystal  forces  and  to  enter  the  vapor  phase 
as  gaseous  molecules.  That  is,  every  crystal  is  able  to  evaporate 
and  has  at  each  temperature  a definite  vapor  pressure  which  is 
the  pressure  at  which  the  rate  of  escape  of  the  molecules  from  the 
crystal  forces  is  just  balanced  by  the  rate  at  which  the  gas  mole- 
cules which  are  constantly  colliding  with  the  crystal  are  in  turn 
caught  and  held  again  by  these  same  forces,  thus  producing  a 
condition  of  dynamic  equilibrium. 

If  we  gradually  impart  heat  to  the  crystal,  its  temperature 
rises  and  the  rate  at  which  molecules  escape  from  the  influence 
of  the  crystal  forces  increases.  Finally  a temperature  will  be 
reached  where  the  crystal  breaks  up  by  the  above  process  faster 
than  it  can  reform  and,  if  the  pressure  is  great  enough,  the  liquid 
phase  appears,  that  is,  the  crystal  is  said  to  melt  or  to  fuse.  The 
temperature  at  which  this  occurs  is  the  melting  point  of  the 
substance.  It  is  evident  that  if  the  addition  of  heat  is  now  con- 
tinued, no  further  rise  in  temperature  will  occur  until  all  the 
crystalline  phase  has  disappeared  since  the  more  violently  oscil- 
lating molecules  in  the  crystal  escape  to  form  liquid  molecules 
and  thus  the  average  amplitude  of  those  remaining,  and  hence 
4 also  the  temperature  of  the  crystal,  remains  constant,  the  heat 

being  absorbed  by  the  process  of  fusion. 

2.  Heat  of  Fusion. — The  heat  absorbed  in  the  process  of  fusion 
evidently  goes  to  increase  the  amplitude  of  oscillation  of  the  mole- 
cules in  the  crystal  to  such  a value  that  they  can  escape  from  the 
field  of  action  of  the  crystal  forces.  In  other  words,  energy  is 
required  to  separate  them  against  the  action  of  these  forces.  A 
i partial  mechanical  analogy  to  this  process  occurs  when  we  whirl 

a ball  at  the  end  of  an  elastic  cord.  As  we  increase  its  energy  of 
rotation,  and  hence  its  angular  velocity,  the  cord  stretches  and 
6 


76 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  VII 


finally  when  a certain  velocity  is  attained  the  cord  breaks  and 
the  ball  flies  off  in  a straight  line.  The  heat  absorbed  in  thfe 
process  of  fusion  is  called  the  latent  heat  of  fusion  of  the  sub- 
stance and  the  molal  heat  of  fusion,  LF,  of  any  substance  at 
the  temperature,  T,  is  defined  as  the  quantity  of  heat  which  is 
absorbed  when  one  mole  of  the  substance  changes  from  the 
crystalline  state  at  the  temperature,  T,  to  the  liquid  state  at 
the  same  temperature.  It  is  exactly  equal  to  the  quantity  of 
heat  evolved  when  the  reverse  process  (crystallization)  occur§ 
at  the  same  temperature. 

Problem  2. — Suppose  we  start  with  a crystal  at  a low  temperature,  and 
at  constant  pressure  gradually  add  heat  to  it,  measuring  the  quantity  of 
heat  added  for  each  degree  rise  in  temperature  which  it  produces,  and  con- 
tinuing the  process  until  we  have  our  substance  in  the  form  of  vapor  at  a 
temperature  100°  above  its  boiling  point.  If  we  construct  a diagram  in 
which  quantities  of  heat  are  plotted  as  abscissae  and  corresponding  rises  in 
temperature  as  ordinates,  two  different  diagrams  will  be  obtained  according 
to  whether  the  melting  point  of  the  substance  is  above  or  below  its  subli- 
mation point  at  the  pressure  employed  in  the  experiment.  Draw  two 
figures  illustrating  the  character  of  the  diagrams  obtained  in  the  two  cases. 
For  simplicity  it  may  be  assumed  that  the  substance  is  not  polymorphic. 

Problem  3. — If  the  substance  is  polymorphic  with  a transition  tempera- 
ture a few  degrees  below  the  melting  point,  what  will  be  the  characters  of 
the  above  curves? 

3.  Supercooled  Liquids  and  Amorphous  Solids. — The  growth 
of  crystal  nuclei  in  a liquid  which  has  been  cooled  below  its  freez- 
ing point  is  facilitated  by  the  presence  of  fine  dust  particles,  by 
mechanical  agitation,  and  by  other  factors.  In  the  absence  of 
these  aids  it  is  frequently  possible  to  cool  a liquid  considerably 
below  its  freezing  point  before  crystallization  occurs.  In  fact 
supercooling,  as  it  is  called,  is  the  usual  phenomenon.  A super- 
cooled liquid  can  usually  be  caused  to  crystallize  by  adding  some 
of  its  own  crystals  to  it.  As  soon  as  crystallization  begins  the 
temperature  very  quickly  rises  to  the  freezing  point  owing  to  the 
heat  evolved  by  the  process  of  crystallization. 

In  all  cases  a decrease  in  the  temperature  of  a liquid  is  accom- 
panied by  an  increase  in  its  viscosity  and  with  some  liquids  the 
viscosity  at  the  freezing  point  is  very  large.  As  the  liquid  is 
supercooled  the  viscosity  increases  still  further.  Now  high 
viscosity  means  decreased  freedom  of  molecular  motion  within 


Sec.  3] 


CRYSTAL  LIQUID  SYSTEMS 


77 


the  liquid  and  molecular  motion  is  necessary  for  the  formation 
and  growth  of  the  crystal  nuclei.  If  this  is  prevented  by  a high 
and  steadily  increasing  viscosity,  the  liquid  may  be  cooled  any 
distance  below  its  freezing  point  without  any  crystallization 
occurring  and  the  viscosity  may  finally  become  so  great  that  the 
liquid  becomes  a solid  glass.  It  is  still  in  the,  liquid  state  of 
aggregation,  however,  for  no  second  phase  has  appeared  during 
the  process  of  cooling.  Ordinary  glass  is  a liquid  of  this  char- 
acter. These  supercooled  liquids  are  sometimes  called  amor- 
phous solids  and  are  distinguished  from  crystalline  solids  by  the 
absence  both  of  crystalline  structure  and  of  a definite  melting 
point.  On  heating  they  gradually  soften  until  they  become  quite 
fluid,  but  there  is  no  one  temperature  above  which  they  may  be 
called  liquid  and  below,  solid,  the  change  in  fluidity  being  a 
perfectly  gradual  one.  Sometimes  old  glass  gradually  and  slowly 
crystallizes.  This  process  is  known  as  devitrification. 

Theoretically  any  liquid  could  be  cooled  to  a solid  glass  with- 
out crystallization  occurring  but  practically  we  have  only  been 
able  to  bring  this  about  with  the  more  viscous  liquids.  Water 
has  been  supercooled  as  much  as  80°  but  was  still  quite  fluid  so 
that  when  crystallization  began  it  proceeded  quite  rapidly.  If  the 
viscosity  of  water  be  increased,  however,  by  dissolving  sugar  in 
it,  the  syrup  can  be  supercooled  until  it  becomes  solid  and  rigid 
like  glass,  without  the  formation  of  any  ice. 

When  a solid  is  formed  by  a chemical  reaction  in  a liquid  or 
gas  which  is  at  a temperature  far  below  the  melting  point  of  the 
solid  thus  produced,  the  latter  frequently  appears  in  the  form  of 
a precipitate  which  under  the  microscope  shows  no  evidence  of 
crystalline  structure.  For  this  reason  these  precipitates  are, 
called  amorphous  precipitates.  It  is  probable,  however,  that  in 
many  instances  these  solids  really  belong  to  the  crystalline  state 
of  aggregation  but  that  the  individual  crystals  are  too  small  to 
be  seen  even  with  the  highest  powers  of  the  microscope. 

Problem  4.— Describe  an  experiment  which  might  be  made  to  determine 
whether  an  apparently  amorphous  precipitate  were  a crystalline  solid  or  a 
supercooled  liquid.  (Cf.  Problem  2.) 


CHAPTER  VIII 


RELATIONS  BETWEEN  PHYSICAL  PROPERTIES  AND 
CHEMICAL  CONSTITUTION 

1.  Nature  of  the  Subject. — Considerable  attention  has  been 
devoted  by  investigators  to  the  question  of  the  connection  be- 
tween the  physical  properties  of  a substance  and  its  chemical 
constitution  and  a large  number  of  relationships  have  been 
proposed  for  expressing  quantitatively  this  connection.  Some 
of  the  physical  properties  which  have  been  the  subject  of  investi- 
gation in  this  field  are  the  following:  surface  tension,  viscosity 
and  fluidity,  atomic  and  molecular  volume,  atomic  and  mole- 
cular heat  capacity,  heats  of  fusion,  of  vaporization  and  of  com- 
bustion, melting  and  boiling  points,  critical  constants,  refrac- 
tivity,  dispersive  powers,  absorption  spectra,  dielectric  constant, 
magnetic  susceptibility  and  permeability,  ionization  constants, 
and  penetrating  power  of  the  characteristic  Rontgen  radiations, 
It  would  not  be  possible  within  the  scope  of  this  book  to  attempt 
to  discuss  or  even  to  present  more  than  a very  small  fraction  of 
the  large  number  of  relationships  which  have  been  put  forward  by 
the  different  investigators  in  this  field,  but  the  general  character 
of  our  present  knowledge  of  this  subject  can  be  well  illustrated 
by  considering  two  or  three  typical  physical  properties  and  the 
manner  in  which  they  depend  upon  the  chemical  constitution 
of  the  substance.  For  this  purpose  the  properties,  optical  ro- 
tatory power,  molecular  refractivity,  and  penetrating  power  of 
the  characteristic  Rontgen  radiation  have  been  chosen.  The 
consideration  of  atomic  and  molecular  heat  capacities  will  be 
taken  up  later  in  a special  chapter  devoted  to  these  properties. 

2.  Optical  Rotatory  Power. — When  monochromatic  light  of 
any  wave  length  is  allowed  to  pass  through  a Nicholas  prism 
(a  special  prism  made  from  Iceland  spar)  the  light  which  emerges 
from  the  prism  is  plane  polarized,  that  is,  the  light  vibrations 
instead  of  occurring  in  all  planes  are  confined  to  a single  plane. 

78 


Sec.  2]  PROPERTIES  AND  CHEMICAL  CONSTITUTION 


79 


If  this  light  is  now  examined  through  a second  Nichol’s  prism 
(called  the  analyser,  the  first  prism  being  known  as  the  polarizer), 
with  its  axis  parallel  to  that  of  the  first  prism,  the  light  is  ap- 
parently not  affected.  If,  however,  one  of  the  prisms  is  now 
gradually  rotated  about  its  axis,  the  observer  notes  a gradual 
decrease  in  the  intensity  of  the  light  as  seen  through  the  analyser. 
This  decrease  continues  until  complete  extinction  is  reached  when 
the  prism  has  been  rotated  through  an  angle  of  90°  from  its  first 
position.  If  now  a solution  of  some  substance  such  as  sugar 
be  placed  between  the  two  prisms  so  that  the  light  passes  through 
the  solution,  the  field  of  view  in  the  analyser  will  become  illu- 
minated again  and  in  order  to  produce  extinction  once  more  the 
analyser  must  be  turned  through  an  angle,  a,  whose  magnitude 
is  a measure  of  the  optical  rotatory  power  of  the  sugar  solution. 
If  it  is  found  that  the  analyser  must  be  turned  to  the  left  in  order 
to  produce  extinction,  the  optically  active  substance  in  the  solu- 
tion is  said  to  be  laevo -rotatory,  while  if  the  rotation  of  the 
analyser  is  to  the  right,  the  substance  is  said  to  be  dextro-rota- 
tory. The  magnitude  of  the  rotation  depends  upon  (1)  the 
wave  length  of  the  light  employed,  being  larger  the  shorter  the 
wave  length;  (2)  the  length  of  the  layer  of  the  solution  through 
which  the  light  passes;  (3)  the  nature  of  the  optically  active 
substance  and  its  concentration  in  the  solution,  and  (4)  the 
temperature. 

The  specific  rotatony  power,  [a],  of  a substance  is  defined  by  the 
.equation, 


it 


100a 

l-p-D 


(i) 


where  a is  the  angular  rotation  observed  when  a solution  contain- 
ing p per  cent,  of  the  substance  and  having  the  density,  D,  is 
examined  through  a tube  of  length,  l,  at  the  temperature,  t. 
The  nature  of  the  monochromatic  light  employed  is  usually 
indicated  by  a subscript.  Thus  [a]^°  indicates  that  the  value 
is  for  20°  C.  and  that  the  Z)-line  of  the  sodium  spectrum  was 
employed  as  the  monochromatic  light.  The  specific  rotatory 
power  of  a substance  is  in  general  a function  both  of  the  tem- 
perature and  of  the  nature  of  the  solvent  in  which  it  is  dissolved. 

The  property  of  optical  activity  is  closely  and  uniquely  con- 


80 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  VIII 


nected  with  the  chemical  structure  of  the  molecule  of  the  optic- 
ally active  substance.  It  occurs  in  the  case  of  every  substance 
whose  molecule  contains  an  asymmetric  central  atom  (see  I, 
2d,  which  should  be  re-read  in  this  connection)  and  also  in 
nearly  all  cases  in  which  the  tetrahedral  space  model  of  the  mole- 
cule possesses  no  plane  of  symmetry.  The  quantitative  side  of 
the  relation  between  optical  activity  and  molecular  structure 
is  not  very  well  developed  at  present  although  the  measurement 
of  the  specific  rotatory  power  of  a substance  can  in  certain  cases 
be  employed  to  determine  the  structure  of  its  molecule  or  to  de- 
cide which  of  two  possible  structures  is  the  correct  one.2 


Fig.  11. 


3.  Molecular  Refractivity. — When  a ray  of  light  passes  from 
one  medium  into  another  (from  air  into  water,  for  example)  the 
ray  undergoes  a change  in  direction,  that  is,  it  is  refracted.  The 
angle  made  by  the  incident  ray  (AB,  Fig.  11)  with  a perpendicu- 
lar at  the  point  of  incidence  is  termed  the  angle  of  incidence, 
i,  while  that,  r,  made  by  the  refracted  ray  (BC  in  the  figure)  is 
termed  the  angle  of  refraction.  The  index  of  refraction,  n,  of 
a substance  is  defined  by  the  equation 

sin  i Ui 
n = . = 

sin  r u% 


(2) 


Sec.  3]  PROPERTIES  AND  CHEMICAL  CONSTITUTION  81 


where  light  of  a single  wave  length  is  supposed  to  enter  the  sub- 
stance from  a vacuum,  the  velocity  of  the  light  in  the  vacuum 
being  ui,  and  that  in  the  substance  being  w2.  In  practical 
work  air  may  usually  be  employed  instead  of  a vacuum.  For 

n2  — 1 1 

a given  substance  the  relation,  n2_^2^D’  w^ere  ^ density, 

has  been  found  to  be  a constant  independent  of  the  temperature. 
This  is  known  as  the  Lorenz  °-Lorentz 6 relation. 

72,2 j AT 

The  molecular  refractivity,  ’ where  ilf  is  the  molec- 


ular weight,  has  been  found  to  be  an  additive  property  in  the 
case  of  many  organic  liquids;  that  is,  to  each  element  a definite 
atomic  refractivity  can  be  assigned  and  from  these  values  the 
refractivity  of  a compound  of  the  elements  may  be  calculated. 
More  extensive  studies,  however,  show  that  not  only  the  nature 
of  the  elements  in  the  compound  but  the  arrangement  of  the 
atoms  in  the  molecule  must  also  be  taken  into  account.  Thus 
the  molecular  refractivities  of  the  two  compounds, 


H-C=C-C=H3 

and  H H 

I I 

c=c=c 

I I 

H H 


will  not  be  the  same  even  though  they  have  the  same  atomic  com- 
position. Account  must  be  taken  of  the  fact  that  one  molecule 
contains  single  and  triple  bonded  carbon  atoms  while  the  other 
contains  only  single  and  double  bonded  carbon.  Further  in- 
vestigations have  shown  that  the  positions  of  the  atoms  in  the 
molecule  with  respect  to  one  another  and  to  double  or  triple 
bonds  also  exert  an  influence  upon  their  atomic  refractivities, 
so  that  the  whole  question  becomes  a very  complicated  one. 
When  employed  with  care,  however,  the  measurement  of  the 
refractive  index  of  an  organic  liquid  may  be  and  frequently  has 

° L.  Lorenz,  Formerly  in  the  University  of  Copenhagen. 

6 Hendrick  Anton  Lorentz,  Professor  of  Physics  in  the  University  of 
Leyden. 


82 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  VIII 


been  of  considerable  value  in  determining  its  chemical  constitution. 
This  same  statement  holds  in  varying  degrees  with  reference  to 
many  other  physical  properties. 

The  extent  to  which  a given  physical  property  of  an  element 
in  a compound  is  affected  by  the  natures  of  the  other  elements 
with  which  it  is  combined  and  by  the  manner  in  which  its  atoms 
are  united  with  those  of  the  other  elements  to  form  the  mole- 
cule of  the  compound  will  obviously  depend  upon  what  character- 
istic of  the  element  is  chiefly  operative  in  determining  the  magni- 
tude of  the  property  in  question.  Thus,  if  the  property  is  one 
whose  magnitude  depends  upon  the  shape  or  the  size  of  the 
atom  or  upon  the  nature  of  its  motion  in  space,  or  if  it  depends 
upon  the  positions,  number,  or  movements  of  the  electrons  in 
or  near  the  surface  of  the  atom,  then  it  is  not  difficult  to  under- 
stand why  the  property  in  question  will  be  affected  by  chemical 
constitution,  for  each  of  the  factors  mentioned  might  easily  be 
affected  by  the  influences  of  the  neighboring  atoms  in  the  mole- 
cule. Only  in  the  case  of  physical  properties  which  are  practi- 
cally entirely  determined  by  conditions  within  the  core  of  the 
atom  and  thus  removed  from  the  influence  of  external  condi- 
tions surrounding  the  atom,  might  we  expect  to  find  properties 
which  are  not  affected  by  such  factors  as  chemical  constitution, 
pressure,  temperature  and  state  of  aggregation.  Mass  is,  of 
course,  the  principal  and  almost  the  only  strictly  additive  prop- 
erty, that  is,  the  total  mass  of  any  compound  is  always  the  sum 
of  the  masses  of  the  elements  which  entered  into  reaction  to  form 
the  compound,  irrespective  of  the  structure  or  composition  of  this 
compound.  Nearly  all  other  physical  properties  are  constitu- 
tive properties  to  a greater  or  less  degree  depending  upon  the 
nature  of  the  property.  A recently  discovered  property,  how- 
ever, seems  to  resemble  mass  in  being  unaffected  by  any  external 
conditions  surrounding  the  atom.  We  shall  consider  this  prop- 
erty briefly. 

4.  Penetrating  Power  of  the  Characteristic  Rontgen  Radia- 
tion of  an  Element. — When  Rontgen®  rays  (X-rays)  are  allowed 
to  fall  upon  a substance  the  substance  in  turn  is  caused  to  emit 

° William  Conrad  Rontgen,  F.  R.  S.  (1845-  ).  Professor  of  Experi- 

mental Physics  in  the  University  of  Munich.  The  discoverer  of  X-  or  Ront- 
gen Rays. 


Sec.  4]  PROPERTIES  AND  CHEMICAL  CONSTITUTION 


83 


secondary  Rontgen  rays  whose  penetrating  power  seems  to  be 
determined  solely  by  the  nature  of  the  elements  in  the  emitting 
substance.  In  the  accompanying  table  (Table  XIII)  are  shown 
the  results  of  a series  of  measurements  of  the  penetrating  power 
of  the  characteristic  Rontgen  radiation  of  the  element  bromine 
in  different  compounds.  The  character  of  the  emitted  rays 

Table  XIII 


Illustrating  the  penetrating  power  of  the  characteristic  Rontgen  radia- 
tion of  the  element  bromine  in  different  chemical  compounds  and  in  different 
states  of  aggregation  (Chapman,  Phil.  Mag.,  21,  449  (1911)). 


Previous  per 
cent,  absorption 
by  A1 

Per  cent,  absorption  by  A1  (0.0062  cm.  thick) 

Radiation  from 

C2H6Br  vapor 

NaBr  solid 

BrOH  solid 

0 

24.1 

24.8 

24.0 

24 

24.4 

24.3 

24.5 

42 

24.1 

23.4 

24.2 

75 

24.7 

23.6 

evidently  appears  to  be  quite  unaffected  by  the  chemical  or  phys- 
ical condition  of  the  element.  Similar  measurements  with  the 
element  iron  show  that  its  characteristic  Rontgen  radiation  is 
the  same  at  room  temperature  as  it  is  at  red  heat  and  is  identical 
from  ferrous  and  from  ferric  compounds.  The  penetrating 
power  of  this  characteristic  radiation  increases  gradually  and  con- 
tinuously with  increasing  atomic  weight  of  the  emitting  element 
being  too  small  to  be  measured  in  the  case  of  the  elements  with 
atomic  weights  less  than  24.  Unlike  most  of  the  physical  prop- 
erties of  the  elements  this  property  is,  therefore,  not  a periodic 
function  of  the  atomic  weights.  Its  complete  independence  of 
external  conditions  surrounding  the  atoms  indicates  strongly 
that  it  is  closely  connected  with  the  nature  of  the  cores  of  the 
atoms  giving  rise  to  it. 

Within  the  last  year  (1913-14)  a more  exact  study  of  the  nature 
of  characteristic  Rongten  radiations  has  become  possible  through 
the  employment  of  crystals  as  spectrometer  gratings,  (V,  6) 
and  in  this  way  Moseley3  has  found  that  the  spectrum  of  the 
characteristic  radiations  is  extremely  simple,  being  composed 
of  very  few  lines.  The  frequencies  corresponding  to  these  lines, 
moreover,  are  closely  connected  with  the  positions  of  the  elements 


84 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  VIII 


in  the  periodic  system  and  investigations  in  this  field  are  giving 
us  a new  insight  into  the  significance  of  the  periodic  system  and 
the  relations  of  the  elements  to  one  another.4  These  relationships 
will  receive  detailed  consideration  in  a later  chapter. 

REFERENCES 

Books:  (1)  The  Relation  between  Chemical  Constitution  and  Physical 
Properties.  Samuel  Smiles,  1910. 

Journal  Articles:  (2)  Hudson,  Jour.  Amer.  Chem.  Soc.,  32,  388  (1910). 
(3)  Moseley,  Phil.  Mag.,  27,  703  (1913).  (4)  Rydberg,  Ibid,  28,  144  (1914). 


CHAPTER  IX 


THE  BROWNIAN  MOVEMENT  AND  MOLECULAR 
MAGNITUDES 

1.  The  Brownian  Movement. — In  order  to  account  for  the 
known  behavior  of  material  bodies,  they  were  early  assumed 
(I,  1)  to  be  made  up  of  very  small  particles  called  molecules 
which  were  in  a state  of  very  rapid  and  constant  unordered 
motion.  The  examination  of  a pure  substance,  a drop  of  liquid 
for  example,  with  the  highest  powers  of  the  best  modern  micro- 
scope fails  to  reveal  the  presence  of  any  such  rapidly  moving 
particles,  however,  and  hence,  if  the  liquid  is  made  up  of  such 
particles,  they  must  be  so  small  as  to  be  beyond  the  range  of  our 
most  powerful  microscopes.  How  could  the  presence  of  these 
rapidly  moving  molecules  be  rendered  visible? 

Suppose  we  were  to  stir  into  a liquid  some  insoluble  substance 
in  an  exceedingly  fine  state  of  division,  the  particles  of  which  were 
in  fact  so  small  that  their  presence  in  the  liquid  could  barely  be 
detected  with  the  microscope.  Now  if  the  liquid  is  in  reality 
composed  of  molecules  moving  to  and  fro  in  all  directions  with 
the  enormous  velocities  assigned  to  them  b}^  the  molecular  theory 
(see  II,  prob.  2),  then  it  is  clear  that  the  collisions  of  the  rapidly 
moving  molecules  with  these  small  visible  particles  (called  col- 
loidal particles)  ought  to  set  the  latter  into  motion  also  and 
through  a properly  arranged  microscopic  system  one  should  be 
able  to  observe  and  study  the  motion  of  these  colloidal  par- 
ticles. The  presence  of  visible  particles  possessing  an  irregular 
motion  was  noticed  in  1827  by  Robert  Brown,  an  English  bot- 
anist while  examining  with  the  microscope  a liquid  containing 
some  pollen  grains.  The  cause  of  this  irregular  motion,  called 
from  its  discoverer  the  Brownian  Movement,  was  not.  suspected 
until  a number  of  years  later,  however.  The  researches2  of 
Wiener  (1863),  Ramsay  (1876),  Delsaulx  and  Carbonelle  (1880), 
Gouy  (1888),  and  others  demonstrated  that  the  movement  could 

85 


80 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IX 


not  be  due  to  convection  currents  in  the  liquid,  that  it  was  nearly 
independent  of  the  nature  of  the  colloidal  particles,  that  it  was 
more  rapid  the  smaller  the  particles  and  the  less  viscous  the  liquid, 
and  that  it  was  persistent  and  never  changing,  continuing  day 
and  night,  month  after  month.  It  has  in  fact  been  observed  in 
small  quantities  of  liquid  found  in  little  pockets  in  granite  and 
other  rocks  where  it  must  have  been  shut  up  for  millions  of  years. 
All  of  these  facts  pointed  to  the  theory  first  suggested  by  Wiener 
that  the  Brownian  Movement  is  the  result  of  molecular  motion 
within  the  liquid.  In  other  words  the  small  visible  colloidal 
particles  are  knocked  about  by  colliding  with  the  invisible  mole- 
cules like  foot  balls  in  the  midst  of  a crowd  of  invisible  players. 

2.  The  Distribution  of  Colloidal  Particles  under  the  Influence 
of  Gravity. — A liquid  containing  colloidal  particles  is  called  a 
“colloidal  solution’’  and  if  such  a solution  is  kept  undisturbed 
at  a constant  temperature  for  some  time,  the  colloidal  particles 
are  found  to  be  distributed  so  that  the  density  of  their  distribu- 
tion is  greatest  near  the  bottom  of  the  vessel  and  decreases  with 
height.  This  is  a similar  behavior  to  that  shown  by  the  atmos- 
phere as  one  rises  above  the  surface  of  the  earth  and  Einstein0 
showed  that,  if  the  BroWnian  Movement  is  caused  by  molecular 
motion,  the  distribution  of  the  particles  with  height  must  follow 
quantitatively  the  law  which  governs  the  decrease  in  density 
of  the  atmosphere  with  height.  1 his  law  can  be  readily  derived 
from  the  perfect  gas  law  and  may  be  expressed  as  follows: 


, Dx  Mgfc  - hx) 
l0g'  IK  = RT 


(1) 


where  D i and  D2  are  the  densities  of  air  at  the  heights  hi  and  h2 
respectively,  M is  the  molecular  weight  of  air  (II,  12),  g the 
acceleration  due  to  gravity,  R the  gas  constant,  and  T the  abso- 
lute temperature. 

Problem  1. — Derive  the  above  equation  from  the  perfect  gas  law  assuming 
T to  be  independent  of  h. 


For  the  molecular  weight  M we  can  put 

M = mN  (2) 


° Albert  Einstein,  since  1914,  Professor  of  Theoretical  Physics  at  the 
University  of  Berlin.  Formerly  at  the  University  of  Zurich. 


Art.  2] 


THE  BROWNIAN  MOVEMENT 


87 


where  m is  the  mass  of  one  molecule  (or  colloidal  particle)  and 
N is  Avogadro’s  number  (I,  6)  and  in  the  case  of  the  colloidal 
solution,  for  m we  can  put, 

m = VD  = V(DC  - Di)  (3) 

where  V is  the  volume  of  the  colloidal 
particle,  and  D,  its  “density  in  the 
solution,”  is  equal  to  its  absolute 

density,  Dc,  minus  the  absolute 

density,  Dh  of  the  liquid  in  which  it 
is  suspended.  We  thus  obtain  the 
relation 

, Di  . nx 

l°ge  D = lOge  ~ = 

“ NF(Z)C  — Di)g{hz  - hx) 


RT 


(4) 


where  nx  and  n2  are  the  average  num- 
ber of  colloidal  particles  in  any  given 
volume  at  the  heights  hi  and  h2  re- 
spectively. 

Perrin®  was  able  to  determine  the 
numbers,  ni  and  n2,  by  photography 
(see  Fig.  12)  and  by  direct  count,  and 
the  difference  in  level,  h2  — h1}  could 
be  read  directly  from  the  micrometer 
screw  of  the  microscope.  Since  for 
a given  colloidal  solution  the  other 
quantities  in  equation  (4)  are  con- 
stants, Perrin  was  thus  able  to  make 
a quantitative  test  of  the  equation. 

In  this  way  he  found  in  one  experi- 
ment2 that  the  number  of  particles 
at  four  different  levels  were  in  the 
ratios,  116,  146,  170  and  200,  while 
the  values  calculated  from  equation 
(4)  far  the  same  levels  were  119,  142,  169  and  201.  Numerous 
other  experiments  gave  similar  results  thus  confirming  quanti- 

a Jean  Perrin,  Professor  of  Physical  Chemistry  at  the  Sorbonne,  Paris. 


Fig.  12. — Micrographs  of 
a colloidal  solution  of  mas- 
tic at  three  different  levels 
12/z  apart.  (Perrin,  J.  Phys., 
Jan.  1910,  p.  24.) 


88 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IX 


tatively  the  theory  that  the  Brownian  Movement  is  the  result 
of  molecular  impacts. 

But  Perrin  was  able  to  go  further  than  this.  By  means  of 
fractional  centrifugation  he  was  able  to  prepare  uniform  colloidal 
particles  of  gamboge  and  of  mastic  of  any  desired  size.  He  was 
also  able  to  measure  the  diameter  and  hence  the  volume  of  the 
individual  particles  by  several  different  methods  which  gaye 
concordant  results.  Knowing  the  volume  of  the  particles  it  is 
clear  that  equation  (4)  can  be  employed  to  calculate  a value  for 
N,  Avogadro’s  number.  For  this  purpose  Perrin  conducted  a 
set  of  experiments  in  which  the  volume  of  the  particles  employed 
was  varied  fifty-fold,  the  density  of  the  suspending  liquid 
fifteen-fold,  and  its  Viscosity  250  fold  in  the  different  experiments. 


B 


Fig.  13. — Path  of  a colloidal  Particle.  (Reproduced  from  Les  Atomes 

by  J.  Perrin.) 

From  all  of  these  experiments,  however,  substantially  the  sajne 
value,  6 — 7 X 1023,  was  obtained  for  N and  this  is  the  value  which 
is  found  for  this  constant  by  a number  of  entirely  different 
methods,  again  furnishing  a strong  quantitative  confirmation 
of  the  theory  of  the  cause  of  the  Brownian  Movement. 

3.  The  Law  of  the  Brownian  Movejnent. — If  a given  colloidal 
particle  be  watched  under  the  microscope  it  will  be  found  to  move 


Sec.  3] 


THE  BROWNIAN  MO  YEMEN T 


89 


through  a very  irregular  and  complicated  path.  Fig.  13  shows 
the  horizontal  projection  of  the  path  of  a colloidal  particle  which 
occupied  the  position  A at  the  beginning  of  the  observation  and 
which  had  reached  the  position  B,  t minutes  later,  having  mean- 
while occupied  successively  all  the  positions  indicated  by  the  small 
dots.  The  distance  from  A to  B in  a direct  lipe  is  called  the 
horizontal  displacement,  X,  of  the  particle  in  the  time  t,  and  is 
determined  by  the  energy  of  agitation  of  the  particle  and  the 
resistance  offered  to  its  motion  by  the  viscosity  of  the  suspending 
liquid.  Einstein  was  the  first  to  derive  the  quantitative  expres- 
sion for  the  displacement  of  a colloidal  particle.  On  the  assump- 
tion that  the  Brownian  Movement  was  the  result  of  the  impacts 
of  the  molecules  of  the  liquid  with  the  colloidal  particles  he 
showed1  that  for  a large  number  of  observations  the  average 
value  of  the  square  of  the  horizontal  displacement  ( X 2)  of  a spher- 
ical colloidal  particle  in  the  time  t ought  to  be 


X2  = 


RT  t 
N Sir rrj 


(5) 


where  r is  the  radius  of  the  particle  and  rj  the  viscosity  of  the 
medium.  In  deducing  this  relation  Einstein  also  assumed  that 
the  motion  of  the  particle  under  a constant  force  / took  place 
in  accordance  with  Stokes’®  Law,  that 


u 


f 

(jTrrrj 


(6) 


where  u is  the  velocity  of  the  particle. 

Equation  (5)  was  tested  by  Perrin  using  his  colloidal  solutions 
of  gamboge  and  mastic.  He  was  able  to  show  that  the  motion 
of  the  particles  obeyed  Stokes’  Law  and  that  the  proportionality 
between  X2  and  t required  by  the  equation  of  Einstein  was  also 
fulfilled.  From  his  measured  values  of  X2  he  computed  the  value 
of  N by  means  of  Einstein’s  equation  with  the  following  very 
striking  results:1 

° George  Gabriel  Stokes,  Kt.,  F.  R.  S.  (1819-1903).  Lucasian  Professor 
of  Mathematics  at  Cambridge  University.  Author  of  many  important 
contributions  to  hydrodynamics  and  optics. 


90 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IX 


Character  of  solution 

i 

Radius  of 
the  parti-  I 
cles  in  ju  ; 

Mass  of  1 
the  par-  | 
tides 
raXlO15 

Number  of  1 
observed 
displace- 
ments 

N X10-23 

0.50 

600 

100 

8.0 

Gamboge  in  water 

0.367 

246 

1500 

6.9 

0.212 

48 

900 

6.9 

Gamboge  in  a 35  per  cent,  solu- 

0.212 

48 

400 

5.5 

tion  of  sugar  (temperature 
poorly  defined). 

Mastic  in  water 

0.52 

650 

1000 

7.3 

Mastic  in  a 27  per  cent,  solu- 

5.50 

750,000 

100 

7.8 

tion  of  urea. 

Gamboge  in  a 10  per  cent,  gly- 

0.385 

290 

100 

6.4 

cerine  solution. 

It  will  be  noticed  from  the  table  that  the  equation  of  Einstein 
holds  fairly  well  even  for  such  large  variations  in  mass  as  15,000- 
fold.  Perrin  considers  the  value  6.9  • 1023  to  be  his  most  reliable 
one. 

More  recent  investigations  by  Nordlund  in  the  laboratory  of 
Svedberg®  at  Upsala  give  a somewhat  lower  value.  Nordlund 
employed  a colloidal  solution  of  mercury,  the  perfectly  spherical 
particles  of  which  had  a radius  of  3 /x.  His  apparatus  was  ar- 
ranged so  that  the  motion  of  the  particles  was  recorded  auto- 
matically upon  a moving  photographic  film,  thus  eliminating  any 
errors  due  to  personal  judgment  concerning  the  path  of  the  par- 
ticle under  observation.  Nordlund’s  experiments  showed  that 
the  value  of  X2  was  directly  proportional  to  the  time,  as  required 
by  Einstein’s  equation  (see  Fig.  14)  and  he  also  found  by  inde- 
pendent experiments  that  the  tiny  spheres  of  mercury  obeyed 
Stokes’  law.  Nordlund’s  experiments  are  probably  the  most 
trustworthy  ones  upon  which  to  base  a calculation  of  N from  the 
Brownian  Movement  in  liquids.  He  found5  as  a mean  value  from 
twelve  experiments  N = 5.9  X 1023,  the  average  deviation  of 

« Theodor  Svedberg,  Professor  of  Physical  Chemistry  at  the  University 
of  Upsala. 


Sec.  4] 


THE  BROWNIAN  MOVEMENT 


91 


the  individual  values  from  this  mean  being  10  per  cent.  It  seems 
therefore  safe  to  conclude  that  from  the  study  of  the  Brownian 
Movement  in  liquids,  we  find  the  value  of  Avogadro’s  Number 
to  be  N = (5.9  • 1023  ± 10  per  cent.). 

4.  The  Brownian  Movement  in  Gases. — Owing  to  the  compara- 
tive simplicity  of  conditions  in  the  gaseous  state  (XI,  3a)  the 
determination  of  the  value  of  N from  the  study  of  the  Brownian 
i Movement  in  gases  offers  fewer  difficulties  than  in  the  case  of 

liquids.  Because  of  the  greater  distances  separating  the  mole- 
cules of  a gas,  the  collisions  between  the  colloidal  particles  and 
the  molecules  are  less  frequent  and  the  mean  free  paths  (III,  2) 


123456  789  10 

Time  in  34  Seconds  — > 


Fig.  14. — Illustrating  the  proportionality  between  time  and  the  square 
of  the  horizontal  displacement  of  a colloidal  particle,  as  required  by  Ein- 
stein’s equation.  (Nordlund,  Z.  physik.  Chem.,  87,  59  1914.) 

of  the  colloidal  particles  are,  therefore,  much  longer  than  in 
liquids.  Thus  the  average  displacement  of  a given  colloidal 
particle  in  a given  time  is  increased  130 -fold  by  transferring  the 
particle  from  water  to  air  at  1 mm.  pressure. 

The  most  accurate  study  of  the  Brownian  Movement  in  gases 
has  been  carried  out  by  Millikan  and  Fletcher  at  the  University 
of  .Chicago.  They  employed  a tiny  drop  of  oil  as  the  colloidal 
particle.  Such  a drop  when  watched  through  a telescope  is 
seen  to  dart  rapidly  about  to  and  fro  in  all  directions  and  forms 

7 


92 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  IX 


a very  vivid  picture  of  the  motion  of  the  invisible  gas  molecules 
through  collision  with  which  it  is  knocked  about  in  this  rapid  and 
irregular  fashion.  From  a large  number  (5900)  of  measurements 
of  the  displacements  along  one  axis  of  12  such  particles,  Fletcher 
has  very  recently7  obtained  the  value,  N = (6.03  X 1023  ± 2 per 
cent.)  for  Avogadro’s  constant. 

The  accurate  value  (6.062  • 1023  ± 0.25  per  cent.)  for  N given 
in  Chapter  I was  obtained  by  Millikan®  by  an  entirely  different 
experimental  method  which  will  be  described  in  a later  chapter. 
It  will  be  noticed  that  the  two  most  reliable  values  for  N yielded 
by  the  Brownian  Movement  method  agree  with  the  above  more 
exact  value  within  the  experimental  error  of  their  own  deter- 
mination. 

Problem  2. — The  smallest  colloidal  particle  which  can  be  detected  under 
the  ultramicroscope  has  a diameter  of  about  3 /jl/jl.  How  many  molecules 
are  there  in  a spherical  colloidal  particle  of  gold  having  this  diameter?  How 
many  in  one  of  benzene?  Densities  19.5  and  0.9. 

Problem  3. — Assuming  the  colloidal  particles  of  gold  3 /ufj,  in  diameter 
to  be  the  “molecules”  of  a gas,  what  would  the  “molecular  weight”  of  this 
gas  be?  What  would  be  the  molecular  velocity,  in  miles  per  second,  of 
these  molecules?  The  specific  gravity  of  gold  is  19.5.  (Cf.  II,  prob.  2.) 

Problem  4. — Calculate  the  number  of  mercury  molecules  in  one  cubic 
centimeter  of  a Torricellian  vacuum  at  20°.  (See  Table  IX,  Chap.  IV.) 

Problem  5. — Calculate  the  vapor  pressure  of  a substance  whose  saturated 
vapor  contains  only  one  billion  molecules  per  cubic  centimeter. 

REFERENCES 

Books:  (1)  Les  Atomes,  J.  Perrin,  1914.  Translated  into  German  by 
A.  Lottermoser.  (2)  The  Brownian  Movement  and  Molecular  Reality. 
J.  Perrin.  Translated  by  F.  Soddy,  1911.  (3)  Die  Existenz  der  Molecule. 

Th.  Svedberg,  1912. 

Journal  Articles:  (4)  New  Proofs  of  the  Kinetic  Theory  of  Matter  and 
the  Atomic  Theory  of  Electricity.  Millikan,  Popular  Science  Monthly, 
Apr.  and  May,  1912.  (5)  Nordlund,  Z.  physik.  Chem.,  87,  60  (1914).  (6) 

Millikan,  Phys.  Rev.,  1,  220  (1913).  (7)  Fletcher,  Ibid.,  Nov.  or  Dec.  (1914). 

a Robert  Andrews  Millikan  (1868-  ).  Professor  of  Physics  in  the 

University  of  Chicago. 


CHAPTER  X 


SOME  PRINCIPLES  RELATING  TO  ENERGY 

I 

1.  Energy. — Energy  as  a concept  may  be  defined  as  the  agency 
postulated  by  science  as  the  underlying  cause  of  all  changes  which 
we  observe  in  the  properties,  or  condition  of  any  portion  of  the 
material  universe.  In  accordance  with  the  conditions  under 
which  it  manifests  its  presence,  energy  is  usually  classified  under 
the  following  forms:  kinetic  energy,  gravitation  energy,  cohesion 
energy,  disgregation  energy,  electrical  energy,  magnetic  energy, 
chemical  energy,  radioactive  energy,  heat  energy,  radiant  energy, 
etc.  We  shall  not  stop  at  this  point  to  explain  just  what  is  meant 
by  each  of  these  terms  as  the  explanations  can  be  more  conveni- 
ently given  from  time  to  time  as  we  shall  have  occasion  to  employ 
the  terms.  A word  of  explanation  is  necessary,  however,  with 
regard  to  the  term  potential  energy  which  we  shall  have  occa- 
sion to  employ  in  connection  with  the  kinetic  energy  of  molecular 
motion. 

Potential  energy  is  usually  defined  as  the  energy  which  an 
elastic  body  or  system  possesses  by  virtue  of  its  configuration, 
that  is,  by  virtue  of  the  relative  positions  of  its  component  parts. 
Thus,  if  we  apply  a force  to  any  system  in  such  a way  as  to 
gradually  and  continuously  change  its  configuration,  using  only 
the  force  necessary  to  bring  about  the  desired  change,  we  are 
said  to  do  work  upon  the  system.  If  the  system  is  a perfectly 
elastic  one,  that  is,  if  it  will  of  itself  on  removal  of  the  impressed 
force  return  completely  to  a configuration  which  as  far  as  its 
energy  is  concerned  is  identical  with  its  original  configuration, 
then  it  is  capable,  in  so  returning,  of  doing  an  amount  of  work 
exactly  equal  to  that  expended  in  changing  it  from  its  first  to  its 
second  configuration.  In  the  second  configuration,  therefore, 
it  is  said  to  possess  potential  energy  because  it  has  a tendency 
of  itself  to  change  to  another  configuration  as  soon  as  the  re- 
straint upon  it  is  removed,  and  in  so  changing  is  capable  of  per- 

93 


94 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


forming  a definite  amount  of  work.  Its  potential  energy  in 
configuration  2 is  quantitatively  defined  as  equal  to  the  work 
which  the  system  is  capable  of  performing  in  changing  from 
configuration  2 to  configuration  1. 

The  following  examples  will  serve  to  illustrate  the  concept  of 
potential  energy.  A steel  rod  bent  or  twisted  under  tension,  a 
stretched  rubber  string,  and  a compressed  fluid,  are  all  examples 
of  systems  possessing  potential  energy.  A system  composed  of 
the  earth  and  a stone  held  at  any  position  above  the  surface  of 
the  earth  also  possesses  potential  energy,  for,  if  the  stone  is 
allowed  to  fall,  it  is  capable  of  doing  a definite  amount  of  work 
before  it  reaches  the  surface  of  the  earth.  If  a stone  of  mass,  m, 
be  thrown  directly  upward  by  the  action,  of  some  force,  it  starts 
with  an  initial  velocity  ui,  and,  therefore,  possesses  an  initial 
kinetic  energy,  \ mui2.  As  it  rises  its  velocity  steadily  decreases 
until  it  finally  becomes  zero.  Its  kinetic  energy  also  becomes 
zero  at  the  same  time.  The  system  composed  of  the  stone  and 
the  earth  is,  however,  now  said  to  possess  an  amount  of  potential 
energy  exactly  equal  to  the  initial  kinetic  energy  of  the  stone,  for 
in  falling  to  the  earth  the  stone  will  again  acquire  the  kinetic 
energy,  \ mu2.  Loss  of  kinetic  energy  in  this  case  is  accompanied 
by  the  gain  of  an  exactly  equivalent  quantity  of  potential  energy, 
the  sum  of  the  two  remaining  always  the  same.  More  generally 
stated,  the  capacity  of  the  system  for  performing  work  is  always 
the  same  whatever  be  the  position  of  the  stone,  until  it  has 
returned  once  more  to  the  surface  of  the  earth  and  its  kinetic 
energy  is  converted  into  heat. 

2.  The  Nature  of  Heat  Energy. — If  two  perfectly  elastic  bodies, 
two  molecules  of  a monatomic  perfect  gas  such  as  helium  for 
example,  collide  with  each  other,  the  total  kinetic  energy  pos- 
sessed by  the  two  bodies  is  not  changed  by  the  collision.  When 
a monatomic  perfect  gas  is  heated,  therefore,  the  heat  which  it 
absorbs  should  all  be  used  up  in  increasing  the  kinetic  energy  of 
translatory  motion  of  the  molecules  of  the  gas.  This  conclusion 
is  in  perfect  accord  with  the  experimental  facts  concerning  the 
specific  heat  of  monatomic  gases.  The  total  heat  content  of  a 
mass  of  a monatomic  perfect  gas  containing  n molecules  would, 
therefore,  be  \nmu2,  where  \mu2  is  the  mean  translatory  kinetic 
energy  of  the  molecules. 


Sec.  2]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


95 


If  the  gas  is  di-,  tri-,  or  polyatomic,  however,  this  would  not 
necessarily  be  the  case,  for  in  a collision  between  two  such  mole- 
cules some  of  the  energy  of  the  collision  might  go  to  increasing 
the  distances  separating  the  atoms  within  the  molecule  and 
would  thus  be  doing  work  against  the  forces  holding  the  atoms 
in  their  positions  or  orbits.  The  corresponding  increase  in  the 
intra-molecular  energy  would  be  classed  as  an  increase  in  intra- 
molecular 'potential  energy  because  it  is  regarded  as  existing  by 
virtue  of  the  new  configuration  of  the  molecule.  An  increase  in 
intra-molecular  kinetic  energy  might  also  occur  because  some  of 
the  energy  of  the  collision  would  probably  go  to  increasing  the 
vibration  or  oscillation  of  the  atoms  within  the  molecule.  Sipii- 
larly  when  heat  is  absorbed  by  a liquid,  a crystal,  or  a com- 
pressed gas,  part  of  the  energy  may  be  used  up  in  doing  work 
against  internal  forces  acting  between  molecules  or  between 
atoms  within  molecules.  It  therefore  becomes  inter-  or  intra- 
molecular potential  energy.  When  heat  is  abstracted  from  such 
a body  by  thermal  conduction  this  inter-  and  intra-molecular 
potential  energy  is  first  transformed  into  unordered  molecular 
kinetic  energy  and  is  then  given  up  to  the  surroundings.  Hence 
as  far  as  our  purposes  are  concerned  this  molecular  potential 
energy  acts  like  a reservoir  of  molecular  kinetic  energy  since  it 
is  not  accessible  to  us  except  by  previous  transformation  into 
unordered  molecular  kinetic  energy.1  Viewed  from  the  stand- 
point of  its  availability  for  purposes  of  doing  useful  work,  there- 
fore, any  form  of  energy  may  be,  and  in  fact  must  be,  classed 
with  heat  energy  if  it  is  of  such  a nature  that  we  are  not  able, 
even  in  principle,  to  abstract  it  from  a system  until  it  has  first 
been  converted  into  unordered  molecular  kinetic  energy  or 
unordered  vibratory  radiant  energy. 

The  quantity  of  heat  energy  which  must  be  imparted  to  one 
gram  of  any  pure  substance  in  order  to  raise  its  temperature 
one  degree  without  change  of  state  is  called  the  specific  heat 
capacity  of  the  substance  and  the  product  of  the  specific  heat  into 
the  molecular  or  atomic  weight  is  called  the  molal  (or  molecular) 
heat  capacity  or  the  atomic  heat  capacity.  The  subject  of  specific 
heats  will  be  considered  more  in  detail  in  a later  chapter. 

1 Or  unordered  vibratory  motion  in  the  ether  in  case  it  is  transmitted  from 
the  body  by  radiation. 


96 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


3.  The  First  Law  of  Thermodynamics. — We  have  just  seen 
that  the  heat  content  of  a body  or  system  of  bodies  arises  from 
the  kinetic  and  potential  energy  possessed  by  its  moving  atoms 
and  molecules.  In  addition  to  its  content  of  heat  energy  a sys- 
tem may,  of  course,  contain  energy  in  other  forms  (chemical  or 
electrical,  for  example)  and  the  total  amount  of  all  forms  of 
energy  which  any  system  contains  is  called  its  total  energy  or 
its  internal  energy  and  is  represented  by  the  letter  U.  We 
know  nothing  as  to  the  magnitude  of  the  total  energy  of  any 
system  but  when  any  change  takes  place  in  a system  there  is 
usually  a corresponding  change  in  its  total  energy,  either  a de- 
crease due  to  its  giving  up  some  of  its  energy  to  the  surround- 
ings or  an  increase  through  receiving  energy  from  the  surround- 
ings. These  changes  in  the  total  energy  of  a system  we  are  able 
to  study  and  to  measure,  and  experience  has  shown  that  they 
obey  the  following  law,  known  as  the  law  of  the  conservation  of 
energy  or  the  First  Law  of  Thermodynamics : When  a quan- 
tity of  energy  disappears  at  any  place  a precisely  equivalent 
quantity  appears  at  some  other  place  or  places;  and  when  a 
quantity  of  energy  disappears  in  any  form  a precisely  equivalent 
quantity  simultaneously  appears  in  some  other  form  or  forms. 
In  other  words  energy  can  neither  be  created  nor  destroyed.  Two 
quantities  of  energy  are  said  to  be  equivalent  if,  when  converted  into 
the  same  form  (heat,  for  example),  they  yield  identical  amounts 
of  that  form.  Thus  if  a certain  amount  of  radiant  energy  be 
entirely  absorbed  by  1 gram  of  water  with  the  result  that  the 
temperature  of  the  water  rises  1°,  the  amount  of  radiant  energy  is 
equivalent  to  1 calorie  of  heat  energy.  If  a certain  quantity 
of  electricity  flows  through  a coil  of  wire  immersed  in  one  gram 
of  water  and  thereby  produces  a rise  of  one  degree  in  the  tem- 
perature of  the  water,  the  electrical  energy  is  equivalent  to  one 
calorie  of  heat  energy,  and  the  quantities  of  radiant  energy  and 
of  electrical  energy  involved  in  these  two  experiments  are, 
therefore,  equivalent  to  each  other. 

It  has  been  found  convenient  to  divide  the  change  in  total 
energy  which  accompanies  any  process  taking  place  within  a 
system  into  two  classes,  designated  as  heat,  Q,  and  work,  IF, 
respectively.  The  reason  for  this  division  lies  in  a practical 
difference  between  heat  energy,  on  the  one  hand,  and  all  other 


Sec.  3]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


97 


forms  of  energy  on  the  other  hand,  which  difference  is  the  cause 
of  a certain  restriction  upon  our  ability  to  transform  heat  energy 
into  other  forms  of  energy.  This  difference  is  due  to  the  fact 
that  in  the  case  of  heat  energy  (and  in  this  class  are  included 
certain  forms  of  radiant  energy)  the  motion  of  the  moving  parts 
is  unordered,  or  random  motion,  while  in  the  case  of  all  other 
forms  of  energy  involved  in  any  process  of  energy  transformation 
or  transference  the  movement  of  the  parts  whose  motion  gives 
rise  to  the  energy  in  question  is  ordered  motion,  that  is,  it  is 
directed  in  one  or  two  or  at  most  a few  definite  directions,  instead 
of  having  the  random  character  of  molecular  motion. 

On  the  basis  of  the  above  classification  of  energy  it  follows 
from  the  conservation  law  that  when  any  process  takes  place 
within  a system,  the  corresponding  change  in  the  total  energy 
of  the  system  must  be  made  up  of  the  work  of  the  process  and 
the  heat  of  the  process,  or  more  exactly  stated,  the  increase, 
A U,  in  the  total  energy,  U,  of  any  system,  which  occurs  when  any 
process  or  change  takes  place  within  the  system,  is  equal  to  the 
quantity  of  heat,  Q,  absorbed  by  the  system  from  the  surround- 
ings, diminished  by  the  quantity  of  work,  W,  done  by  the  system 
upon  the  surroundings  , or  in  mathematical  language. 

AU  = Q — W (1) 

This  equation  is  the  mathematical  formulation  of  the  First  Law 
of  Thermodynamics. 

The  following  example  is  an  illustration  of  its  application  to 
a specific  process:  If  a mixture  of  hydrogen  and  oxygen  in  a 
cylinder  provided  with  a weighted  piston  is  exploded,  the  force 
of  the  explosion  will  raise  the  weighted  piston  through  a certain 
distance  against  the  force  of  gravity  and  there  will  be  a simul- 
taneous evolution  of  heat  which  will  be  gradually  taken  up  by 
the  surroundings.  Suppose  the  weight  lifted  to  be  10,000  grams 
and  that  it  is  raised  10  meters.  The  work  done  upon  it  would 
be  10,000X1000X980  ergs,  where  980  (dynes)  is  the  accelera- 
tion due  to  gravity.  This  is  9,800,000,000  ergs  or  980  joules 
and  is  evidently  work  done  upon  the  surroundings  by  the  sys- 
tem since  it  is  done  against  a force,  gravitation,  exerted  by  the 
surroundings  upon  the  weighted  piston.  Since  one  calorie  is 
equivalent  to  4.2  joules,  the  above  quantity  of  work  is  equivalent 


98 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


to  42  =233  calories.  Suppose  that  the  heat  evolved  by  the 

explosion  is  1000  calories.  The  quantity,  Q,  therefore,  in  equa- 
tion (1)  would  be  —1000  calories,  since  Q has  been  so  defined  as 
to  be  positive  when  heat  is  absorbed  by  the  system  from  the 
surroundings.  For  the  increase,  A U,  in  the  total  energy  of  the 
system  which  results  from  the  above  process  we  have,  therefore, 

AXJ  — Q — W = —1000  — 233=  —1233  cal. 

or,  in  words,  the  total  energy  of  the  system  has  decreased  by 
1233  calories,  since  the  value  of  AU  comes  out  with  a negative 
sign.  The  above  example  of  the  application  of  the  equation  of 
the  First  Law  brings  out  the  fact,  which  the  student  should  always 
keep  in  mind,  that  before  two  energy  quantities  can  be  employed 
together  in  an  equation  they  must  first  be  expressed  in  terms  of  the 
same  energy  unit. 

4.  Corollaries  of  the  First  Law. — A little  consideration  will 
show  that  the  following  two  statements  are  necessary  conse- 
quences of  the  First  Law  of  Thermodynamics : 

1.  The  total  energy  of  any  system  in  a given  state  or  condition 
is,  for  the  system  in  question,  a definite  characteristic  of  that 
state  and  is  independent  of  the  manner  in  which  the  system 
reached  that  state. 

2.  When  a system  changes  from  a state,  A,  where  its  total 
energy  is  U a,  to  some  other  state,  B,  where  its  total  energy  is 
UB,  the  increase  in  total  energy,  A U = Ub~Ua,  which  accompanies 
this  change  is  independent  of  the  process  by  which  the  change  is 
brought  about. 

For  example,  one  gram  of  the  substance,  water,  in  the  form  of 
ice  at  a temperature  of  —10°  and  a pressure  of  one  atmosphere 
contains  a definite  amount  of  energy  which  is  the  same  for  every 
gram  of  water  in  this  condition  irrespective  of  the  previous  his- 
tory of  the  water  provided  it  has  been  in  this  condition  for 
sufficient  time  to  reach  a state  of  equilibrium.  If  we  wished  to 
convert  a gram  of  water  in  the  above  condition  into  one  gram  of 
water  vapor  at  a temperature  of  200°  and  a pressure  of  one-tenth 
of  an  atmosphere,  we  could  do  so  in  a variety  of  different  ways. 
For  example,  we  might  first  reduce  the  pressure  upon  the  ice 
until  it  had  all  evaporated  and  then  the  vapor  could  be  heated 


Sec.  5]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


99 


to  200°  and  the  pressure  brought  to  one-tenth  of  an  atmosphere; 
or,  we  might  first  melt  the  ice,  heat  the  resulting  water  to  boil- 
ing, boil  it  all  away  and  then  heat  the  steam  to  200°  and  bring 
the  pressure  to  one-tenth  of  an  atmosphere;  or,  we  could  dissolve 
the  ice  in  an  aqueous  solution  of  sulphuric  acid  and  pass  a current 
of  electricity  through  this  solution  between  platinum  electrodes, 
until  one  gram  of  a mixture  of  hydrogen  and  oxygen  gases  were 
evolved.  This  mixture  could  then  be  exploded  and  the  water 
vapor  formed  could  be  cooled  to  200°  and  the  pressure  upon  it 
brought  to  one-tenth  of  an  atmosphere.  By  whatever  process 
the  above  change  is  brought  about,  however,  we  should  find  that 
the  total  energy  change  accompanying  the  process  is  always  the 
same  irrespective  of  the  nature  of  the  process. 

5.  Work  and  Energy  Units. — The  work  associated  with  any 
process  may  always  be  regarded  as  ordered  motion  taking  place 
under  the  influence  of  a force,  /,  and  when  so  regarded,  is  quan- 
titatively defined  as  the  product  of  the  force  into  the  distance 
through  which  it  acts.  On  the  basis  of  this  definition  it  can  be 
readily  shown  that  in  the  case  of  a change  in  the  volume  of  a 
system  against  a pressure,  the  work  done  by  the  system  is  equal 
to  the  product  of  the  increase  in  volume,  Av,  and  the  pressure,  p, 
under  which  it  takes  place  or,  for  an  infinitesmal  volume  increase, 
we  have 


dW  = pdv  (2) 

If  the  pressure  is  constant  during  the  change  in  volume,  the  in- 
tegral of  this  expression  is 

W = p(v2  — Vi)  = pAv  (3) 

where  Vi  and  v2  are  the  initial  and  final  volumes,  respectively. 
If  the  pressure  is  expressed  in  atmospheres  and  the  volumes  in 
liters,  the  corresponding  energy  unit  is  called  the  liter-atmosphere, 
one  liter-atmosphere  being  defined  as  the  work  done  when  a 
volume  increase  of  one  liter  occurs  against  a constant  pressure 
of  one  atmosphere. 

In  the  case  of  electrical  work  our  definitions  of  electrical 
quantities  are  such  that  the  electrical  work  done  when  a current 
of  electricity  flows  under  a difference  of  potential  is  equal  to  the 
quantity,  q,  of  electricity  multiplied  by  the  difference  of  poten- 


100 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


tial,  E,  under  which  it  flows.  If  the  quantity  of  electricity  is 
expressed  in  coulombs  and  the  potential  difference  in  volts,  the 
product  will  be  volt-coulombs  or  joules. 

Work,  heat,  or  any  form  of  energy  may,  of  course,  be  expressed 
in  any  one  of  the  various  energy  units,  ergs,  joules,  calories,  or 
liter-atmospheres.  The  definitions  of  the  various  units  have 
already  been  given  in  the  Introduction.  They  are  related  to 
one  another  quantitatively  by  the  following  equations: 


1 liter-atmosphere 


= 24.207  cal. 

= 4. 184  joules 


1 cal. 


1 joule  (by  definition)  = 107  ergs. 


6.  The  Relation  Connecting  Heat  of  Fusion,  Heat  of  Vapor- 
ization, and  Heat  of  Sublimation. — 

Problem  1. — Prove  that  the  First  Law  of  Thermodynamics  requires  that 
the  molal  heat  of  fusion  (Lp),  of  vaporization — of  the  liquid — ( Lv ),  and  of 
sublimation  (Ls),  in  the  case  of  any  substance  shall  be  connected  by  the 
equation, 


(4) 


provided  all  of  the  processes  mentioned  occur  at  the  same  constant  tempera- 
ture, T,  and  pressure,  P. 

Problem  2. — In  order  to  melt  one  gram  of  ice  at  0°  and  a pressure  of  one 
atmosphere  79.60  calories  of  heat  are  required.  To  convert  1 gram  of  water 
under  the  same  conditions  into  saturated  vapor  requires  2494.6  joules  of 
energy.  How  many  calories  of  heat  will  be  evolved  during  the  deposition 
of  18  grams  of  hoar  frost  from  saturated  air  at  0°? 

7.  The  Second  Law  of  Thermodynamics. — With  respect  to 
the  transformation  of  work  into  heat  (that  is,  of  ordered  or 
directed  motion  into  unordered  or  random  molecular  motion) 
our  experience  teaches  us  that  there  is  no  restriction  except  that 
contained  in  the  First  Law  of  Thermodynamics  which  merely 
requires  that  in  such  a transformation  the  amount  of  heat  pro- 
duced shall  be. exactly  equivalent  to  the  work  expended,  or  in 
other  words  that  there  shall  be  neither  destruction  of  energy  nor 
creation  of  energy  out  of  nothing.  But  with  respect  to  the 
reverse  transformation  of  heat  into  work  (of  random  molecular 
motion  into  directed  motion  of  some  kind),  our  experience  teaches 
us  that  there  are  certain  practical  restrictions  imposed  by  nature 
in  addition  to  those  contained  in  the  statement  of  the  First  Law. 


Sec.  7]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


101 


The  complete  statement  and  description  of  the  nature  of  these 
restrictions  comprises  wfat  is  known  as  the  Second  Law  of 
Thermodynamics. 

That  some  additional  restriction  is  perhaps  to  be  expected, 
knowing  as  we  dp^rSie  random  nature  of  the  motion  which  gives 
rise  to  heat  energy,  will  be  appreciated  from  the  following  con- 
siderations: . In  the  case  of  a moving  mass  larg6  enough  for  us 
to  control  of  *as  an  individual  we  can  theoretically  convert  all 
or  any  of  its  kinetic  energy  into  some  other  form  of  useful 
energy,  or  in  other  words,  can  make  it  do  any  desired  form  of  work. 
In  the  case  of  a moving  molecule,  however,  the  moving  body  is 
too  small  for  us  to  control  its  motion  as  an  individual.  We 
can  see,  therefore,  at  once  that,  as  far  as  we  are  concerned,  the 
kinetic  energy  of  such  a small  particle  as  a molecule  is  not  avail- 
able for  our  uses  in  the  same  sense  as  that  of  a body  large  enough 
for  us  to  control  as  an  individual.  Suppose,  however,  that  we 
have  a very  large  number  of  small  particles  all  moving  in  the 
same  direction,  or  of  whose  motion  there  is  a sufficiently  large 
component  in  some  one  direction.  We  have  such  a situation, 
for  example,  in  a metallic  wire  through  which  a current  of  elec- 
tricity is  passing.  The  current  consists  of  a stream  of  electrons 
(I,  2g)  which  at  any  moment  are  all  moving  in  one  direction 
through  a given  cross  section  of  the  wire,  or  at  least  the  motion 
of  every  electron  has  a positive  component  in  this  direction.  In 
such  a case  the  particles  are  likewise  too  small  to  be  controlled 
as  individuals  but  they  have  a resultant  kinetic  energy  in  one 
direction  which  is  large  enough  for  us  to  control  and  we  are  thus 
able  to  transform  this  energy  into  some  desired  form  of  useful 
work. 

In  the  case  of  heat  energy,  however,  we  are  dealing  with  a very 
large  number  of  particles,  too  small  to  be  controlled  as  individu- 
als, and  whose  motion  is  perfectly  at  random;  that  is,  if  we  were 
to  resolve  the  motion  of  all  the  molecules  of  a gas  or  a liquid  in 
any  direction  whatever,  the  resultant  motion  and  hence  also 
the  resultant  kinetic  energy  in  this  direction  for  any  finite  inter- 
val of  time  would  be  zero,  as  far  as  our  purposes  are  concerned. 
In  order,  therefore,  to  convert  unordered  molecular  kinetic 
energy  into  ordered  kinetic  energy,  that  is,  in  order  to  trans- 
form heat  into  work,  we  are  obliged  to  slow  up  all  of  the  moving 


102 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


molecules,  or  in  other  words,  to  reduce  their  average  kinetic 
energy  and  hence  to  lower  the  temperature  of  the  body  from 
which  we  wish  to  take  the  heat  energy. 

For  example,  when  a gas  expands  and  raises  a weight  it  does 
work,  but  in  doing  so  it  always  cools  off,  that  is,  the  average 
kinetic  energy  of  its  molecules  decreases,  the  total  decrease  in 
the  molecular  and  atomic  kinetic  and  potential  energy  being 
exactly  equivalent  to  the  work  done.  If  the  gas  is  kept  in  a 
large  heat  reservoir  during  its  expansion,  it  will  take  up  heat  from 
the  reservoir  which  in  turn  will  cool  down.  That  is,  the  heat 
energy  necessary  for  performing  the  desired  work  will  come  from 
the  reservoir  instead  of  from  the  gas  itself,  and,  if  the  heat  reser- 
voir is  large  enough  in  comparison  with  the  amount  of  gas  em- 
ployed, the  amount  by  which  it  is  cooled  down  will  be  infinitesimal, 
since  the  decrease  in  molecular  energy  will  be  distributed  over 
such  an  enormous  number  of  molecules  that  the  average  kinetic 
energy  of  each  molecule  and  hence  the  temperature  of  the  reser- 
voir would  remain  practically  unaltered.  For  practical  purposes, 
therefore,  we  may  regard  a compressed  gas  in  good  thermal  con- 
tact with  a sufficiently  large  heat  reservoir  (such  as  the  ocean, 
for  example)  as  a device  for  converting  the  heat  energy  of  the 
reservoir  into  useful  work  at  practically  constant  temperature 
and  as  long  as  our  supply  of  compressed  gas  lasted  we  might  go 
on  converting  heat  into  work  in  this  manner.  Large  supplies  of 
compressed  gas  are  not  available,  however,  and  if  we  attempted 
to  compress  the  gas  after  expansion  with  the  idea  of  using  it  over 
again,  we  would  find  that,  even  under  the  most  theoretically 
perfect  conditions,  it  could  not  be  compressed  to  its  original 
pressure  except  by  the  expenditure  of  at  least  as  much  work  as 
could  be  obtained  from  it  during  the  expansion  described  above. 
Practically  we  would  always  find  that  more  work  would  be 
required.  In  other  words  the  operation  could  not  be  worked  in 
a cycle  for  the  purpose  of  transforming  heat  into  work.  In  fact 
all  of  our  experience  leads  to  the  conclusion  that: 

A system  or  arrangement  of  matter  operating  in  a cycle  can- 
not transform  heat  into  work  in  surroundings  of  constant  tem- 
perature. This  statement  evidently  constitutes  a restriction 
upon  the  transformation  of  heat  into  work  and  is  part  of  the 
Second  Law  of  Thermodynamics.  The  rest  of  the  law  has  to  do 


Sec  7]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


103 


with  the  transformation  of  heat  into  work  by  a machine  operating 
between  two  temperatures.. 

If  we  place  a body  having  the  temperature,  Th  in  thermal  con- 
tact with  a colder  body  having  the  temperature,  T 2,  we  invariably 
find  that  heat  flows  from  the  hot  body  to  the  cold  body  and  never 
in  the  reverse  direction.  That  is,  heat  energy  will  never  of  itself 
flow  from  a lower  to  a higher  temperature  but  only  in  the  reverse 
direction.  This  result  can  be  shown  to  be  a necessary  consequence 
of  the  laws  of  mechanics  applied  to  a mechanical  system  composed 
of  moving  masses  such  as  the  molecules  of  a body  and  from  the 
kinetic  point  of  view  is  equivalent  to  the  statement  that,  if  a 
system  composed  of  a large  number  of  moving  masses,  such  as  the 
molecules  of  a perfect  gas,  having  the  average  kinetic  energy, 
\ mui2,  is  brought  into  contact  with  a second  similar  system  hav- 
ing the  average  kinetic  energy,  \ mu22,  in  such  a manner  that  a 
distribution  of  momentum  and  of  kinetic  energy  can  occur,  the 
resultant  transfer  of  kinetic  energy  will  necessarily  be  in  the 
direction  of  the  system  having  initially  the  smaller  value  of 
§ mu2.  Whenever,  therefore,  two  bodies  of  matter  at  different 
temperatures  are  brought  into  thermal  contact  in  any  way  there 
will  occur  a transfer  of  molecular  kinetic  energy  (i.e.,  a flow  of 
heat)  from  the  hotter  to  the  colder  body.  In  this  flow  of  heat 
just  as  in  the  case  of  the  moving  electrons  described  above,  we 
recognize  once  more,  the  transfer  of  a finite  amount  of  ordered  or 
directed  kinetic  energy  in  one  direction  and  there  exists,  therefore, 
just  as  in  the  case  of  the  electric  current,  the  possibility  of 
transforming  this  directed  energy  into  some  form  of  useful  work.2 

2 The  statement  that  heat  will,  of  itself,  flow  only  from  a higher  to  a lower 
temperature  and  never  in  the  reverse  direction  is  one  of  the  several  methods 
of  expressing  the  basic  principle  upon  which  the  Second  Law  of  The  rmody- 
namics  rests.  This  basic  principle  regarding  the  uni-directional  autogenous, 
flow  of  heat  is,  moreover,  usually  regarded  as  essentially  a new  principle, 
that  is,  one  which  cannot  be  derived  from  the  law  of  conservation  of  energy 
and  the  ordinary  principles  of  mechanics.  This  view,  however,  becomes 
unnecessary  if  we  accept  the  kinetic  interpretation  of  the  meaning  of  heat 
energy.  That  is,  if  we  regard  the  heat  energy  of  a material  body  or  system 
as  the  energy  which  it  possesses  by  virtue  of  the  unordered  motion  of  its 
atoms  and  molecules,  then  it  follows  from  the  ordinary  principles  of  mechanics 
that  the  transfer  of  heat  energy  can  only  take  place  from  a higher  to  a lower 
temperature.  The  criterion  by  which  we  determine  which  of  two  bodies 
or  systems  has  the  higher  temperature  then  becomes  the  following:  Let 


104 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


This  being  the  case,  the  question  naturally  arises  as  to  how  much 
ordered  or  directed  kinetic  energy  is  associated  with  the  passage  of 
a quantity  of  heat  from  a higher  to  a lower  temperature,  for  it  is 
this  ordered  energy  only  which  is  available  for  transformation 
into  useful  work.3  In  order  to  discover  this  it  will  only  be  neces- 
sary to  carry  out  a couple  of  “imaginary  experiments.”  Suppose 
we  have  two  large  heat  reservoirs  each  of  practically  infinite  heat 
capacity.  Let  the  first  reservoir  have  the  temperature,  Ti,  and 
the  second,  the  temperature,  T 2,  and  let  the  two  reservoirs  be 
very  close  together  but  insulated  from  each  other  so  that  no  trans- 
fer of  heat  from  one  to  the  other  occurs.  We  will  suppose, 
further,  that  in  the  interior  of  the  first  reservoir  we  have  a quantity 
of  some  monatomic,  perfect  gas,  such  as  helium  under  low  pres- 
sure, for  example. 

Let  the  following  imaginary  experiments  be  carried  out:  (1) 
Place  the  two  reservoirs  in  thermal  contact  for  such  a length  of 
time  that  a finite  quantity,  Qi,  of  heat  passes  from  the  first  reser- 
voir to  the  second.  The  passage  of  this  heat  may  be  assumed  to 
take  place  either  by  radiation,  or  by  conduction  (along  a metallic 
connecting  rod,  for  example),  or  by  any  other  mechanism  what- 
soever. After  the  passage  of  this  quantity  of  heat  the  two  reser- 
voirs are  again  completely  insulated  from  each  other. 

a quantity  of  a perfect  gas  be  placed  in  thermal  contact  with  the  first  system 
for  a sufficient  time  for  thermal  equilibrium  to  establish  itself.  Call  the 
average  kinetic  energy  of  the  molecules  of  the  gas  under  these  conditions, 
l mup.  Now  take  the  gas  and  place  it  in  thermal  contact  with  the  second 
system  until  thermal  equilibrium  is  established.  Its  average  molecular 
kinetic  energy  is  now  \ w^2.  If  % mup  = \ mu 22,  the  two  systems  are  said 
to  have  the  same  tempe  ature.  If  the  two  kinetic  energies  are  not  equal 
then  the  system  in  which  \ mu'-  has  the  greater  value  is  at  the  higher 
temperature. 

3 The  amount  of  ordered  kinetic  energy  involved  in  the  transfer  of  a given 
quantity  of  heat,  Q,  from  Ti  or  T2  is  a definite  characteristic  of  the  quantity 
Q and  the  two  temperatures  T\  and  T 2 and  is  independent  of  the  manner  in 
which  the  transfer  occurs.  For,  if  this  were  not  the  case,  that  is,  if  the 
amount  of  ordered  kinetic  energy  associated  with  the  passage  of  Q calories 
of  heat  from  Ti  to  by  one  process  of  transfer  were  different  from  that  by 
some  other  process  of  transfer,  it  can  be  readily  proved  that  the  two  processes 
could  be  combined  in  such  a way  as  to  yield  a result  which  would  amount 
simply  to  an  autogenous  flow  of  heat  from  a lower  to  a higher  temperature,  a 
result  which  has  been  shown  above  to  be  impossible. 


Sec.  8]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


105 


Now  as  far  as  the  transfer  of  energy  is  concerned,  the  net  result 
of  the  above  experiment  can  be  exactly  duplicated  by  the  follow- 
ing process:  (2)  Take  from  the  first  reservoir  such  a number,  n , 
of  molecules  of  helium  gas  that  the  total  kinetic  energy,  J nmui2, 
possessed  by  them  shall  be  the  exact  equivalent  of  the  quantity 
of  heat,  Qi,  which  flowed  from  the  first  reservoir  to  the  second,  in 
the  process  just  described.  Transfer  this  quantity  of  gas  bodily 
to  the  second  reservoir  and  allow  it  to  come  into  equilibrium  with 
the  reservoir  keeping  its  volume  constant.  The  gas  will  give  up 
some  of  its  energy  to  the  second  reservoir  and  will  thus  attain  the 
temperature,  T2,  of  this  reservoir,  at  which  temperature  its  mole- 
cules will  have  the  total  kinetic  energy,  | nmu22.  If  the  gas  be 
now  returned  to  the  first  reservoir  again,  the  whole  system  will  be 
in  the  same  condition  as  it  was  at  the  end  of  our  first  experiment. 

It  is  clear  that  the  net  result  of  our  second  experiment  consists 
simply  in  the  transfer  of  the  quantity  of  kinetic  energy,  \ nmu2 
— | nmu2 2,  in  a definite  direction;  that  is,  from  the  first  to  the 
second  reservoir,  from  T i to  T2.  This  represents  ordered  or 
directed  energy  and  is,  therefore,  available  for  conversion  into 
useful  work  with  the  aid  of  some  suitable  mechanism  whose 
nature  we  are  not  at 'present  concerned  with.  Since,  as  explained 
in  footnote  3 on  the  preceding  page,  the  quantity  of  ordered  or 
directed  kinetic  energy  involved  in  the  transfer  of  a given  quan- 
tity, Qi,  of  heat  from  Ti  to  T 2 is  independent  of  the  way  in  which 
the  transfer  occurs,  the  relation 

n(J  mui2—  \ mu22)  = W (5) 

is  a perfectly  general  expression  for  the  available  work  of  such  a 
transfer.  In  this  expression  n represents  that  number  of  mole- 
cules of  a monatomic  perfect  gas  which  at  the  temperature,  Th 
have  a total  kinetic  energy,  J nmui2,  equivalent  to  Q i and  which 
at  the  temperature  T2  have  the  total  kinetic  energy,  | nmu22. 

8.  Carnot’s  Equation. — Equation  (5)  really  represents  the  es- 
sence of  the  Second  Law  of  Thermodynamics.  It  is  customary, 
however,  to  express  the  Second  Law  equation  in  terms  of  quantity 
of  heat  and  temperature  rather  than  in  terms  of  molecular  kinetic 
energy.  Before  translating  equation  (5)  into  these  terms,  how- 
ever, we  will  first  define  the  terms  engine,  heat  engine,  and  per- 
fect engine. 


106 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


Any  system  or  arrangement  of  matter  which  works  in  a cycle 
and  converts  any  form  of  energy  into  some  desired  form  of  useful 
work  will  be  called  an  engine.  The  expression,  “ working  in  a 
cycle,”  means  that  the  working  system  periodically  returns  to 
its  initial  condition.  An  engine  which  works  between  two  differ- 
ent temperatures  and  converts  heat  into  work  will  be  called  a 
heat  engine.  In  all  real  heat  engines  the  efficiency  is  usually 
very  low  owing  to  losses  caused  by  friction  and  by  radiation  and 
conduction,  and  owing  also  to  the  fact  that  in  actual  practice  it 
is  not  energy  which  is  desired  so  much  as  power,  that  is,  energy 
per  second.  As  a result  we  do  not  operate  our  engines  in  practice 
in  such  a way  as  to  obtain  the  maximum  amount  of  work  which 
they  are  capable  of  yielding  for  that  would  mean  that  the  rate  of 
production  of  work  would  be  exceedingly  low.  Instead,  we 
sacrifice  some  of  the  possible  work  for  the  purpose  of  getting 
what  we  do  obtain,  at  a higher  rate  of  speed. 

We  can,  however,  imagine  an  engine  in  which  all  the  parts 
work  so  smoothly  that  there  are  no  friction  losses;  in  which  heat 
insulation  is  so  perfect  that  there  are  no  energy  losses  due  to 
radiation  or  conduction  of  heat;  and  which  is  operated  so  slowly 
that  at  every  moment  it  is  performing  all  the  work  it  is  capable 
of;  that  is,  it  acts  always  against  an  external  force  practically 
equal  to  the  internal  forces  which  are  driving  it.  Such  an 
imaginary  engine  will  be  called  a perfect  engine.  It  evidently 
represents  a limiting  condition  approached  by  all  real  engines  as 
the  losses  mentioned  above  are  reduced  to  a minimum. 

In  the  case  of  a perfect  heat  engine  operating  between  the  two 
temperatures,  T i and  T2,  in  such  a way  that  it  takes  up  Q 1 
calories  of  heat  from  a reservoir  at  the  temperature,  Th  and  trans- 
forms part  of  it  into  work,  the  question  naturally  arises  as  to 
what  fraction  of  the  Qi  calories  of  heat  such  an  engine  is  able  to 
transform  into  work.  This  is  the  question  which  occurred  to  a 
young  Trench  engineer,  Sadi  Carnot,®  about  1824,  and  his  answer 

° Sadi  Nicolas  Leonhard  Carnot  (1796-1832).  The  eldest  son  of  one  of 
Napolean’s  generals;  a captain  of  Engineers  in  the  French  army.  A young 
man  of  remarkably  brilliant  mind;  a profound  and  original  thinker;  the 
founder  of  modern  thermo-dynamics.  His  only  published  work,  Reflexions 
sur  la  puissance  motrice  der  feu  el  sur  les  machines  propres  a developper  cet 
puissance , appeared  in  1824  when  he  was  28  years  of  age.  He  died  of  the 
cholera  in  Paris  at  the  age  of  36. 


Sec.  8]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


107 


to  it  constitutes  one  of  the  common  forms  of  stating  the  Second 
Law  of  Thermodynamics. 

In  order  to  obtain  Carnot’s  equation  from  our  equation  (5) 
we  will  first  multiply  and  divide  the  left  hand  member  by  \nmu\ 2, 
and  obtain 

n(imui2-%mu22)(?nmui2) 

(6) 


Now  it  will  be  recalled  that  \nmu i2  is  the  quantity  of  heat  energy 
possessed  by  the  n molecules  of  helium  gas  at  the  temperature  T i 
and  represents  therefore  the  quantity  of  heat,  Qi,  which  was 
transferred  from  the  first  to  the  second  reservoir  (from  T i to  T2)f 
in  the  process  described  above.  Equation  (6)  may,  therefore, 
be  written 


(imui2  — \mu22) 
\mu i2 


Qi=W 


(7) 


and  since  according  to  equation  (29,  II),  the  mean  kinetic  energy  of 
the  molecules  of  a perfect  gas  is  proportional  to  the  absolute 
temperature  of  the  gas,  our  equation  becomes 

(8) 


or 


eT  i 

Ti—  T2 

t; 


Qi  = W 


Qi  = W 


(9) 


which  represents  Carnot’s  method  of  expressing  what  is  now 
called  the  Second  Law  of  Thermodynamics. 

From  our  method  of  deriving  this  relation  it  is  evident  that  the 
Ti-T2. 

is  that  part  of  the  Q i units  of  heat  which  repre- 


fraction 


T i 


sents  the  amount  of  ordered  or  directed  kinetic  energy  which  is 
transferred  from  T\  to  T 2 and  which  is,  therefore,  available  for 
performing  useful  work.  Stated  in  other  words,  equation  (9) 
signifies  that  any  heat  engine  which  operates  between  two  tem- 
peratures, Ti  and  T2,  in  such  a manner  that  it  takes  up  Q i units 
of  heat  at  the  temperature,  Ti,  and  converts  part  of  it  into  work 
can  never,  even  under  the  most  favorable  conditions  imaginable 
(i.e.,  those  attained  with  a perfect  heat  engine),  convert  more  than 
Ti-T2 

the  fraction,  m , of  this  heat  into  work,  for  this  fraction, 
i i 


8 


108 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


as  we  have  just  seen,  represents  the  total  amount  of  ordered 
kinetic  energy  which  is  involved  in  the  passage  of  Q i units  of  heat 
from  Ti  to  T 2. 

Only  when  T2  equals  zero  would  it  be  theoretically  possible  to 
convert  all  of  the  Qi  units  of  heat  into  work.  In  other  words,  if 
surroundings  at  a temperature  of  absolute  zero  were  available 
for  our  use,  a perfect  heat  engine  could  convert  into  work  all  of 
the  heat  energy  which  it  absorbed  at  any  temperature.  This,  of 
course,  merely  means  that  in  order  to  convert  all  of  the  molecular 
kinetic  energy  of  any  body  into  some  form  of  useful  work  it 
would  be  necessary  to  bring  all  the  molecules  of  the  body  to  rest. 
Since  we  have  no  surroundings  available  at  a temperature  of 
absolute  zero  we  have  to  content  ourselves  with  slowing  up  the 
molecules  of  our  working  system  as  much  as  possible  and  to  this 
end  the  temperature  T\  should  always  be  made  as  low  as  feasible 
and  the  difference,  T\—T2,  as  large  as  feasible. 

The  most  common  form  of  heat  engine  employed  in  practice 
is  the  steam  engine.  The  working  system  is  water  vapor 
enclosed  in  a cylinder  provided  with  a movable  piston  and  the 
heat  energy  is  converted  directly  into  mechanical  work.  The 
two  temperatures  between  which  the  steam  engine  operates  are 
the  boiler  temperature,  Ti,  and  the  temperature  of  the  exhaust, 
T2.  The  boiler  temperature  is  usually  maintained  by  the  com- 
bustion of  some  kind  of  fuel.  The  temperature  of  the  exhaust  is 
not  lower  than  the  temperature  of  the  surroundings. 

In  practice  considerable  quantities  of  heat  energy  escape 
from  the  boiler  to  the  surroundings  by  radiation  and  conduction 
instead  of  being  taken  up  by  the  engine  and  employed  in  doing 
useful  work.  Part  of  the  work  done  by  the  engine  is  also  con- 
verted back  into  heat  again  by  the  friction  of  the  moving  parts  of 
the  engine  and  is,  therefore,  not  available.  Moreover,  as  stated 
above,  engines  are  not  operated  in  practice  so  as  to  perform  at  all 
times  the  maximum  work  which  they  are  capable  of  doing.  As 
the  result  of  all  these  losses  the  efficiency  of  steam  engines 
employed  in  practice  is  much  lower  than  that  of  a perfect  heat 
engine.  Thus,  a perfect  heat  engine  taking  superheated  steam  at 
a temperature  of  206.4°  and  delivering  it  to  a water-cooled  exhaust 

fiQ1o  | | , T1—T2  206.4-43.1 

at  a temperature  ol  43.1  would  convert,  ^ =206  4+273  1 


Sec.  9]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


109 


= 34  per  cent,  of  the  heat  energy  of  the  steam  into  mechanical 
work.  The  most  efficient  1000-H.P.  steam  engine  which  has 
ever  been  constructed, 4 when  working  under  substantially  the  same 
conditions  converts  only  25  per  cent,  of  the  heat  energy  it  receives 
into  useful  work,  while  for  most  steam  engines  as  employed  in 
practice  this  figure  will  average  only  about  8-10  per  cent. 

When  figured  on  the  basis  of  the  total  available  energy  obtain- 
able from  the  coal,  the  efficiency  is  much  smaller  even.  It  can 
be  shown  that  every  gram  of  carbon  which  is  burned  to  CO2 
should  theoretically  be  capable  of  yielding  about  34,000  joules 
of  energy  (at  ordinary  temperatures)  in  the  form  of  useful  work. 
This  amount  of  carbon,  if  burned  under  conditions  analogous  to 
those  under  which  coal  is  burned  in  the  best  modern  boiler  prac- 
tice, will  deliver  to  the  engine  about  27,000  joules  of  energy  in 
the  form  of  heat  and  25  per  cent,  or  6800  joules  of  this  would  be 
converted  into  useful  work  by  the  1000-H.P.  engine  mentioned 
above.  That  is,  the  most  efficient  modern  engineering  methods 

can  obtain  from  coal  only  about,  34QQQ  or  20  per  cent,  of  its 

theoretical  work  producing  power.  The  remaining  80  per  cent, 
is  lost.5  Under  average  working  conditions  only  5-7  per  cent,  of 
the  energy  of  the  coal  is  obtained  in  the  form  of  useful  work.  It 
is  evident,  therefore,  that  modern  methods  of  converting  the 
energy  of  coal  into  useful  work  are  really  exceedingly  wasteful. 

9.  Free  Energy  and  the  Principle  of  the  Degradation  of  Energy. 
— In  addition  to  the  energy  which  is  obtainable  from  the  chemical 
reaction,  C + 02  = C02,  which  represents  the  combustion  of  car- 
bon, many  other  chemical  reactions  are  also  capable  of  yielding 


4 A Nordberg,  air  compressor,  quadruple  expansion  engine  with  regenera- 
tive heating  system.  For  the  tests  on  this  engine  see  Hood,  Trans.  Amer. 
Soc.  Eng.,  1907,  p.  705. 

5 By  employing  mercury  vapor  instead  of  water  vapor  as  the  working 
medium,  the  efficiency  of  a vapor  heat  engine  could  be  increased,  because 
the  boiling  point  of  mercury  (357°)  is  257°  higher  than  that  of  water  and  a 
higher  initial  temperature  (T 1)  could,  therefore,  be  employed.  Recent 
experiments  by  Emmet  (Proc.  Amer.  Inst.  Elec.  Eng.,  33,  473  (1914)) 
indicate  that  a mercury  vapor  engine  connected  in  tandem  with  a steam 
engine  would  show  a 45  per  cent,  increase  in  efficiency  (per  lb.  of  fuel)  over 
the  steam  engine  alone.  An  experimental  engine  of  this  kind  is  under  process 
of  construction. 


110 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


energy  which  can  be  made  available  for  the  production  of  work. 
In  fact,  in  the  case  of  every  process  whether  chemical  or  physical, 
which  tends  to  take  place  of  itself,  a portion  of  the  total  energy 
decrease,  —A U,  which  accompanies  the  process  can  always  be 
obtained  in  the  form  of  useful  work,  if  the  process  is  allowed  to 
take  place  in  a suitable  manner.  The  maximum  amount  of  work 
which  the  process  is  theoretically  capable  of  yielding  when 
carried  out  in  such  a way  as  to  operate  a perfect  engine  of  some 
character  is  called  the  “free  energy,”  A,  of  the  process.  The 
free  energy  of  a chemical  reaction  is  an  important  and  charac- 
teristic property  of  the  reaction. 

In  order  to  make  clearer  the  distinction  between  the  free  energy, 
A,  of  a reaction  and  its  total  energy,  —A U,  we  will  consider  the 
chemical  reaction, 

Zn+Hg2S04+7H20  = ZnS04*7H20  + 2Hg 

which  occurs  when  a rod  of  metallic  zinc  is  placed  in  an  aqueous 
solution  saturated  with  mercurous  sulphate.  In  this  reaction 
the  zinc  dissolves  and  throws  out  the  mercury.  When  allowed 
to  occur  in  this  way,  there  is  evidently  no  work6  done  by  this  reac- 
tion and  the  total  energy,  — AU,  liberated  by  the  reaction  must 
appear  entirely  in  the  form  of  heat.  That  is,  by  the  First  Law 
(equation  (1))  we  have 

— A U=  —Q  = 81,320  cal.  of  heat  evolved. 

If  instead  of  allowing  the  reaction  to  proceed  in  the  above  way, 
we  place  the  mercury  in  the  bottom  of  a tube,  cover  it  with  a paste 
of  mercurous  sulphate  and  then  fill  the  tube  with  a saturated  solu- 
tion of  zinc  sulphate  in  which  the  rod  of  zinc  dips,  we  will  find  on 
connecting  the  zinc  and  the  mercury  to  the  terminals  of  a volt- 
meter that  there  exists  a difference  of  potential  between  them 
amounting  to  1.429  volts  at  18°  and  if  they  be  connected  to  the 
terminals  of  a perfect  electric  motor,  a current  of  electricity 
(193,000  coulombs  of  electricity  for  each  atomic  weight  of  zinc 
dissolved)  will  pass  through  the  motor  during  the  time  the  above 
reaction  is  taking  place,  and  this  current  can  thus  be  made  to  do 

6 With  the  exception  of  the  very  insignificant  amount  of  work  done  against 
the  pressure  of  the  atmosphere  owing  to  the  fact  that  there  is  a slight  change 
in  volume  in  the  above  reaction. 


Sec.  9]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


111 


useful  work.  The  maximum  amount  of  work  which  it  is  capable 
of  doing  represents  the  free  energy  of  the  reaction  and  is  evidently 
given  by  the  expression  (X,  5) 


A = EXq=  1 .429  volts X 193,000  coulombs  = 275,800  joules  = 


275800 

4.184 


= 65,880  calories. 


The  rest  of  the  energy  of  the  reaction  appears  in  the  form  of  heat 
and  amounts  to  —Q^  = 15,440  calories.  We  have,  therefore, 


— AU  = — Q-\-W  = — Q2+A  = 15,440+65,880  = 81,320  calories.  . 

Most  of  the  processes  which  occur  in  nature  and  in  the  indus- 
tries take  place  in  such  a way  that  the  whole  or  the  greater  part 
of  the  energy  which  they  give  out  appears  entirely  in  the  form  of 
heat  which  is  then  radiated  and  conducted  away  to  the  surround- 
ings and  thus  is  no  longer  available  for  the  production  of  work. 
This  constant  transformation  of  available  energy  in  various 
forms  into  unavailable  heat  energy  is  called  the  degradation  of 
energy  and  the  Second  Law  of  Thermodynamics  is  sometimes 
called  the  Principle  of  the  Degradation  of  Energy  because  it 
points  out  that  heat  energy  at  the  temperature  of  our  surround- 
ings is  not  available  for  transformation  into  useful  work.  The 
Principle  of  the  Degradation  of  Energy  is  more  comprehensive 
than  the  Second  Law  of  Thermodynamics  as  embodied  in  the 
Carnot  equation,  however,  for  there  are  other  ways  in  which 
the  capacity  of  a system  for  doing  work  can  diminish  besides  the 
mere  conversion  of  a part  or  the  whole  of  its  available  energy  into 
heat.  For  example,  when  a gas  is  allowed  to  expand  into  a 
vacuum  it  does  no  work.  Some  of  its  work-producing  power  has 
been  lost  by  the  process,  however,  for  it  might  have  been  made  to 
lift  a weight  during  its  expansion.  Many  other  examples  of 
processes  which  are  attended  by  a loss  of  work-producing  power, 
that  is,  a loss  of  “free  energy”  as  it  is  also  called,  might  be  cited 
and  our  experience  teaches  us  that  the  available  energy  of  our 
world  is  constantly  decreasing  owing  to  the  continual  occurrence 
of  such  processes  in  nature  and  in  the  industries.  This  state- 
ment is  the  Principle  of  the  Degradation  of  Energy.  According 
to  the  first  principle  of  energy  (Principle  of  Conservation)  the 


112 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  X 


total  energy  of  the  universe  is  a constant  but  according  to  the 
second  principle  (Principle  of  Degradation)  the  available  energy 
of  the  world  is  constantly  decreasing. 

The  two  laws  of  thermodynamics  are  of  great  importance  in 
physical  chemistry  since  they  enable  us  to  derive  a large  number 
of  important  relations  governing  equilibrium  in  physical  and 
chemical  systems  under  a variety  of  conditions.  We  shall  have 
occasion  to  employ  a number  of  these  thermodynamic  relation- 
ships in  the  following  chapters  but  will  postpone  the  derivations 
of  most  of  them  until  later.  In  the  next  section  an  application 
of  the  Second  Law  to  a system  at  constant  temperature  is  de- 
scribed. 

10.  Proof  of  Equality  of  Vapor  Pressures  at  the  Freezing  Point. — 

Consider  a pure  liquid  at  A (Fig.  15) 
in  contact  at  bb'  with  its  solid  crys- 
tals in  B,  the  space  above  both  the 
liquid  and  solid  being  filled  with 
the  vapor.  Assume  that  both  the 
liquid  and  solid  are  at  the  temper- 
ature of  the  freezing  point  and 
hence  (by  definition)  in  equilibrium 
with  each  other  at  bb'.  If  their 
vapor  pressures  are  not  equal  then 
one  of  them  must  be  greater  than 
the  other.  Suppose  the  vapor 
pressure  of  the  liquid  is  greater 
than  that  of  the  solid.  If  this  be 
the  case  the  liquid  will  evaporate 
at  A and  the  vapor  (which  accord- 
ing to  our  supposition  has  too  high 
a pressure  to  be  in  equilibrium 
with  the  solid)  would  condense  to 
solid  again  at  B,  and  this  process 
of  distillation  would  continue  until 
all  of  the  liquid  had  been  converted  into  solid  which  would  be 
contrary  to  our  assumption  that  they  were  initially  in  equilib- 
rium with  each  other.  The  only  other  possibility  is  that,  as 
fast  as  the  liquid  is  used  up  by  evaporation  at  A,  it  is  replaced 
by  the  melting  of  the  solid  at  bb'.  Under  this  arrangement  we 


a 


Sec.  10]  SOME  PRINCIPLES  RELATING  TO  ENERGY 


113 


would  have  a constant  stream  of  vapor  passing  by  the  position 
aa'  in  the  direction  A— >B  and  by  placing  a turbine  at  this 
point  we  could  obtain  work  from  a system  working  in  a cycle 
at  constant  temperature  which  would  be  contrary  to  the 
Second  Law  of  Thermodynamics  (X,  7).  Hence  the  vapor 
pressure  of  the  liquid  cannot  be  greater  than  that  of  the  solid. 
In  the  same  way  it  can  be  shown  that  the  vapor  pressure  of 
the  solid  cannot  be  greater  than  that  of  the  liquid.  They  must, 
therefore,  be  equal  and  the  freezing  point  is,  therefore,  the  point 
at  which  the  vapor  pressure  curve  of  the  liquid  intersects  that 
of  the  solid.  (See  Fig.  16.) 

By  a similar  method  of  proof  it  can  be  shown  that  if  any  two 
phases  are  both  in  equilibrium  with  a third  phase  they  are  in 
equilibrium  with  each  other. 


CHAPTER  XI 


SOLUTIONS  I:  DEFINITION  OF  TERMS  AND 
CLASSIFICATION  OF  SOLUTIONS 

1.  Definition  of  a Solution. — In  our  first  chapter  (I,  2)  we 
applied  the  term  mixture  to  any  material  which  is  composed  of 
more  than  one  species  of  molecule.  In  this  and  the  following 
chapters  we  shall  deal  with  an  important  class  of  mixtures  known 
as  solutions.  A solution  may  be  defined  as  a one-phase  system 
composed  of  two  or  more  molecular  species  (see  definition  of 
phase,  I,  8).  The  exact  significance  of  this  definition  will  be 
more  easily  understood  from  the  following  considerations. 

If  we  grind  together  sugar,  C12H22O11,  and  sand,  Si02,  we  can 
obtain  an  intimate  mixture  of  these  two  materials,  but  on  close 
examination  we  can  readily  recognize  the  presence  of  two 
crystalline  phases,  namely,  crystals  of  pure  sand  and  crystals  of 
pure  sugar,  in  the  mixture.  This  mixture  is,  therefore,  hetero- 
geneous (I,  8)  and  is  hence  not  a solution.  Similarly,  if  we  shake 
together  (1)  liquid  mercury  and  liquid  benzene,  or  (2)  liquid 
mercury  and  gaseous  nitrogen,  or  (3)  liquid  mercury  and  sugar 
crystals,  we  obtain  in  each  instance  systems  in  which  the  presence 
of  two  phases  is  readily  recognized.  Moreover,  a chemical  exami- 
nation of  the  liquid  and  the  crystalline  phases  in  each  of  the  above 
systems  would  fail  to  give  any  evidence  of  the  presence  of  more 
than  one  molecular  species  within  the  phase,  and  without  such 
evidence  we  could  not  class  any  one  of  the  phases  as  a solution. 

By  bringing  together  (1)  liquid  water  and  liquid  alcohol,  or 
(2)  liquid  water  and  gaseous  hydrochloric  acid,  or  (3)  liquid  water 
and  sugar  crystals,  however,  we  can  obtain  in  each  instance  a 
one-phase  (i.e.,  homogeneous)  system  in  which  the  presence  of 
more  than  one  molecular  species  can  be  readily  ascertained.  We 
obtain,  therefore,  in  each  of  these  cases  a solution. 

114 


Sec.  2] 


SOLUTIONS  I 


115 


In  a true  solution,1  after  equilibrium  is  reached,  the  individual 
molecules  of  the  different  molecular  species  present  are  intimately 
and  uniformly  mixed  with  one  another,  or,  in  another  phraseology 
which  is  frequently  used,  the  different  component  substances 
which  were  brought  together  in  order  to  prepare  the  solution 
have  become  molecularly  dispersed  in  one  another.  The  dif- 
ferent molecular  species  of  which  a solution  is  'composed  will 
be  called  its  molecular  components  and  the  substances  corre- 
sponding to  these  molecular  species,  or  more  generally  the  sub- 
stances of  which  the  solution  is  considered  to  be  composed,  will 
be  called  its  components  or  its  constituents.  For  example,  a 
solution  whose  constituents  are  the  pure  substance,  sugar,  and 
the  associated  substance,  water,  has  the  following  molecular  com- 
ponents: (H2O),  (H2O)  2)  (H2O)  3,  C12H22O11,  C^H^On^H^O)*, 
and  possibly  others. 

2.  Associated  Substances  as  Solutions. — An  associated  sub- 
stance (III,  8)  such  as  water  must,  strictly  speaking,  be  itself 
classed  as  a solution  since  it  contains  more  than  one  species  of 
molecule.  It  is  a solution  of  a peculiar  character,  however, 
because  the  different  molecular  species  all  have  the  formula 
(H20)x,  where  x is  an  integer,  and  these  species  are,  moreover,  all 
in  chemical  equilibrium  (I,  9)  with  one  another,  the  equilibrium 
being  established  so  rapidly  that  we  are  unable  to  separate  any 
one  of  the  molecular  species  from  the  others.  Chemically,  there- 
fore, and  in  many  ways  physically  also,  water  behaves  as  it 
would  if  it  contained  only  the  molecular  species,  H20,  and  is  for 
this  reason  commonly  spoken  of  as  a “pure  substance.”  Re- 
garded as  a solution,  water  is  considered  to  be  made  up  of  the 
substances  hydrol,  H20,  dihydrol,  (H20)2,  trihydrol,  (H20)3, 
etc.,  no  one  of  which,  however,  has  as  yet  been  obtained  in  the 
pure  condition,  at  least  not  as  a liquid.  Owing  to  the  fact  that 
in  so  many  ways  an  equilibrium  mixture,  such  as  water,  resem- 

1 The  term  “true”  solution  is  used  here  because  of  the  existence  of  the 
class  of  systems  to  which  the  name  “colloidal  solutions”  has  been  given. 
These  systems  consist  of  one  phase  very  highly  dispersed  in  another  and 
occupy  a position  intermediate  between  homogeneous  mixtures  (“true” 
solutions)  on  the  one  hand  and  the  ordinary  heterogeneous  systems  on  the 
other.  They  have  already  been  discussed  to  some  extent  in  Chapter  IX 
and  will  be  further  considered  in  a later  chapter. 


116 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XI 


bles  a pure  substance  in  its  behavior,  it  is  convenient  to  class 
such  mixtures  with  pure  substances  for  many  purposes  and  for 
this  reason  the  term  associated  substance,  rather  than  the  term 
solution,  is  usually  applied  to  them.  The  expression  “pure 
water”  can,  therefore,  be  employed,  if  we  agree  to  understand 
thereby  an  equilibrium  mixture  of  molecules  all  of  which  have 
the  formula  (HoO)^,  where  x is  an  integer,  and  in  which  the  equi- 
librium responds  so  rapidly  to  changing  conditions  (temperature, 
pressure,  etc.)  that  it  may  be  regarded  as  establishing  itself 
practically  instantaneously 2 at  each  moment. 

3.  Classification  of  Solutions. — Tor  purposes  of  systematic 
treatment  solutions  are  most  conveniently  classified,  according  to 
their  state  of  aggregation,  as  gaseous  solutions,  liquid  solutions, 
and  crystalline  solutions. 

(a)  Gaseous  Solutions. — Gaseous  solutions  furnish  the  sim- 
plest example  of  solutions,  since  the  molecules  are  here  so  far 
apart  that  they  are  comparatively  without  influence  upon  one 
another.  Owing  to  this  fact  many  of  the  physical  properties  of 
mixtures  of  perfect  gases  are  strictly  additive  (VIII,  3),  that  is, 
a molal  physical  property  for  a mixture  of  gases  may  be  cal- 
culated by  means  of  the  relation, 

2/  = 2/a*a+2/b*b+  ....  2/n*n  (1) 

where  y is  the  molal  property  in  question  ( e.g .,  molal  volume, 
molal  heat  capacity,  or  molal  (molecular)  refractivity)  for  the 
mixture  and  y&,  yB,  etc.,  are  the  corresponding  molal  properties 
for  the  pure  constituents  of  the  mixture,  xB,  etc.,  being  their 
mole  fractions  in  the  mixture.  Equation  (1)  is  an  expression  of 
the  so-called  law  of  mixtures.  It  will  be  noticed  that  Dalton's 
law  of  partial  pressures  (II,  6)  for  gaseous  mixtures  has  the  same 
form  as  the  above  equation. 

Problem  1.- — Two  tubes,  10  mm.  in  diameter,  are  placed  side  by  side  so 
that  white  light  of  a constant  and  uniform  intensity  illuminates  them 
lengthwise.  The  first  tube  is  50  mm.  long  and  is  filled  with  iodine  vapor 

2 Numerous  other  liquids  in  which  there  exists  a chemical  equilibrium 
which  responds  very  rapidly  to  changes  in  external  conditions  behave  in 
many  ways  as  though  they  were  pure  substances,  and  for  many  purposes 
may  advantageously  be  classified  as  such.  The  numerous  tautomeric 
substances  familiar  to  the  organic  chemist  are  good  examples  of  such 
systems. 


Sec.  3] 


SOLUTIONS  I 


117 


(I2)  under  a pressure  of  0.06  atmosphere.  The  second  tube  is  500  mm.  long 
and  is  filled  with  a gaseous  mixture  of  hydrogen  (H2)  and  iodine  (I2)  under 
a pressure  of  0.06  atmosphere.  Both  tubes  are  at  a temperature  of  100°  and 
when  compared  with  each  other  are  found  to  show  the  same  shade  and  in- 
tensity of  violet  color  when  viewed  lengthwise  toward  the  source  of  white 
light.  Calculate  the  concentration  (in  moles  of  I2  per  liter)  of  the  iodine  in 
the  second  tube. 

Owing  to  the  comparatively  large  distances  between  the  mole- 
cules of  gases  and  the  consequent  lack  of  influence  of  one  molecule 
upon  another,  no  energy  change  occurs  when  any  two  perfect 
gases  are  mixed  together  in  the  same  volume,  provided  that  they 
do  not  react  chemically  with  each  other.  This  is  made  evident 
by  the  absence  of  any  temperature  change  when  two  such  gases 
are  mixed  together.  Similarly  when  a perfect  gas  is  allowed  to 
expand  into  a vacuum  there  is  no  change  in  its  temperature. 
These  statements  do  not  hold  for  gases  under  high  pressures, 
however,  for  here  very  pronounced  energy  changes  (heat  effects) 
occur  when  expansion  or  mixing  takes  place.  We  shall  have 
occasion  to  consider  such  energy  changes  in  a later  chapter. 

In  the  case  of  gaseous  mixtures  at  such  high  pressures  or  at 
such  low  temperatures  that  the  perfect  gas  laws  do  not  apply 
with  a sufficient  degree  of  accuracy,  modifications  of  these  laws 
along  lines  similar  to  those  followed  by  van  der  Waals  or  Berthe- 
lot  in  their  treatment  of  pure  gases  (II,  10)  are  usually  employed. 
We  shall  not  consider  them  further  in  this  book. 

(b)  Crystalline  Solutions. — The  subject  of  crystalline  solu- 
tions, or  mixed  crystals,  has  been  briefly  referred  to  in  a previous 
chapter  (V,  4).  Owing  to  the  restricted  nature  of  molecular 
motion  in  crystals  and  the  high  viscosity  of  this  state  of  aggrega- 
tion, crystalline  solutions  in  a state  of  equilibrium  are  seldom  met 
with  in  practice,  because  the  attainment  of  equilibrium  in  a 
reasonable  length  of  time  is  so  frequently  prevented  by  the 
restraints  upon  the  free  movements  of  the  molecules.  (Cf.  V,  3.) 
For  this  reason  crystalline  solutions,  as  usually  met  with  in  prac- 
tice, are  not  amenable  to  the  same  methods  of  treatment  as  are 
nearly  all  gaseous  and  liquid  solutions.  This  same  statement 
frequently  holds  true  also,  and  for  the  same  reason,  for  the  glasses 
(VII,  3),  which  represent  one  type  of  liquid  solutions.  The  dis- 
cussion of  such  cases  of  crystalline  solutions  as  may  be  treated 


118 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XI 


as  systems  in  thermodynamic  equilibrium  will  be  taken  up  in 
connection  with  the  chapter  on  the  Phase  Rule. 

(c)  Liquid  Solutions. — The  most  interesting  group  of  solutions 
to  the  chemist  is  the  class  of  liquid  solutions,  because  by  far  the 
greater  portion  of  the  processes  of  the  chemist  are  carried  out 
in  such  solutions  and  because  they  play  such  a predominant  and 
important  role  in  natural  processes.  For  these  reasons  liquid 
solutions  have  been  and  continue  to  be  the  subject  of  the  most 
careful  study,  and  the  discovery  of  laws  and  the  establishment  of 
satisfactory  theories  for  the  interpretation  of  many  of  the  proc- 
esses occurring  in  such  solutions  has  been  one  of  the  chief  tri- 
umphs of  modern  physical  chemistry.  In  the  following  chapters 
dealing  with  the  subject  of  solutions  we  shall,  therefore,  restrict 
ourselves  to  the  class  of  liquid  solutions. 

4.  The  Constituents  of  a Solution.  Solvent  and  Solute. — 
When  crystals  of  some  substance,  such  as  sugar,  are  treated  with 
water,  the  crystals  are  observed  to  gradually  dissolve  or  disappear 
and  a homogeneous  system  consisting  of  sugar  and  water  is 
eventually  obtained.  Such  a solution  is  commonly  spoken  of 
as  a solution  of  sugar  in  water  and  is  by  some  chemists  regarded 
as  having  been  formed  owing  to  some  specific  solvent  or  dissolving 
action  exerted  by  the  water  upon  the  sugar.  For  this  reason  the 
water  is  commonly  called  the  solvent,  and  the  sugar,  the  dissolved 
substance  or  the  solute.  It  would  be  quite  as  correct,  however, 
to  look  upon  the  solution  as  a solution  of  water  in  sugar  and,  as 
a matter  of  fact,  it  could  be  prepared  by  dissolving  water  in  liquid 
sugar,  if  it  were  desired  to  do  so.  The  sugar  might  then  be  called 
the  solvent  and  water  the  dissolved  substance  or  the  solute. 
Similarly,  if  a little  water  (either  as  a liquid,  or  in  the  form  of  ice 
or  steam)  is  “ dissolved”  in  alcohol,  a solution  of  water  in  alcohol 
is  obtained.  Exactly  the  same  solution  might  be  prepared,  how- 
ever, by  “dissolving”  alcohol  (either  solid,  liquid,  or  gaseous)  in 
water.  The  nature  of  the  solution  thus  obtained  is  entirely 
independent  of  the  method  of  it  preparation,  and  the  designation 
of  one  constituent  as  the  solvent  and  the  other  as  the  solute,  on 
the  above  basis,  is  an  entirely  artificial  and  arbitrary,  not  to  say 
confusing,  distinction.  A better  and  more  general  method  of 
distinguishing  the  constituents  of  a solution  is  simply  to  refer 
to  them  as  constituent  A,  constituent  B,  etc.,  and  this  is  the 


Sec.  5 ] 


SOLUTIONS  I 


119 


method  which  will  be  usually  employed  in  this  book.  The  terms 
solvent  and  solute  will  be  employed  chiefly  in  the  treatment  of 
an  important  class  of  solutions  known  as  “dilute  solutions/’ 
that  is,  solutions  in  which  the  amount  of  one  constituent  is  much 
greater  than  that  of  all  the  other  constituents  together.  For 
such  solutions  the  term  solvent  will  be  employed  to  designate 
the  constituent  which  predominates  in  the  solution,  without 
however  implying  thereb}^  that  this  constituent  exercises  any 
specific  solvent  power  upon  the  others.  The  other  constituents 
will  be  called  the  solutes.  This  method  of  designation  is  in 
accordance  with  common  usage  but  its  purely  conventional 
character  should  not  be  forgotten.3 

The  following  general  treatment  of  the  subject  of  solutions  will 
for  simplicity  be  restricted,  in  most  cases,  to  solutions  made  up 
of  only  two  constituents  which  will  be  designated  either  as  A 
and  B,  or  as  solvent  and  solute,  respectively.  The  relations  and 
laws  which  we  shall  derive  can,  however,  be  readily  extended  to 
solutions  containing  any  number  of  components.  Moreover, 
since  any  substance  A,  in  the  liquid  state,  can  be  regarded  as  a 
limiting  case  of  a solution  of  B in  A in  which  the  amount  of  B 
has  become  zero,  pure  liquids  may  be  treated  as  special  limiting 
cases  under  the  subject  of  solutions  and  will  be  in  certain  cases 
included  in  our  treatment  of  the  subject. 

5.  Methods  for  Expressing  the  Composition  of  Solutions. — 
The  composition  of  a solution  is  frequently  given  in  terms  of 
percentages  (i.e.,  grams  in  100  grams  of  solution)  of  its  various 
constituents.  In  what  follows  we  shall,  however,  usually  express 
the  composition  of  the  solution  in  terms  of  the  mole  fractions 
(II,  6)  of  its  different  components.  Thus,  if  a solution  is 
composed  of  NA  moles  of  A and  AB  moles  of  B,  the  mole  fraction 

of  A will  be  xA  = AT^A  and  that  of  B will  be  £R= 

Aa+Ab 

(II,  6)  and  hence, 


A^a+Ab 


^a+^b  = unity 


(2) 


3 The  common  expression,  “Water  is  the  universal  solvent, ” means  that 
a large  number  of  substances  are  able  to  become  molecularly  dispersed  in 
water  and  that  water  occurs  in  large  quantities  in  nature.  The  ability  of 
two  substances  to  become  molecularly  dispersed  in  each  other  is  a recipro- 
cal relationship,  not  a one-sided  property  as  the  expression  “solvent  power” 
might  be  taken  to  indicate. 


120 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XI 


Problem  2. — Calculate  the  two  mole  fractions  for  each  of  the  following 
solutions:  10  grams  of  water  (H20)  and  3 grams  of  alcohol  (C2H5OH);  8 
grams  of  benzene  (CeHe)  and  1 milligram  of  diphenyl  (Ci2Hi0);  a 10  per 
cent,  solution  of  sugar  (Ci2H220n)  in  water. 

Similarly,  if  we  wish  to  express  the  composition  of  the  solution  in 
terms  of  its  molecular  components  rather  than  its  constituents 
(XI,  1)  the  same  system  will  be  employed  on  the  basis  of  molecular 
7lA  71b 

fractions,  xA=n^_^_n^  xn~ nK-\-n^  etc*>  where  nA , ™b,  etc., 

represent  the  numbers  of  molecules  of  the  molecular  species  A, 
B,  etc.,  which  are  present  in  the  solution.  In  many  cases  the 
mole  fraction  of  a substance  in  a solution  will  be  identical  with  the 
molecular  fraction  of  the  corresponding  molecular  species.  This 
will  of  course  always  be  the  case  when  each  substance  in  the 
solution  has  only  one  molecular  species  corresponding  to  it. 
Thus  in  a solution  composed  of  benzene  and  toluene  the  mole 
fraction  of  the  substance,  benzene  (molal  weight  = 78),  in  the 
solution  is  identical  with  the  molecular  fraction  of  the  molecular 
species,  CeH6;  but  in  a solution  composed  of  water  and  alcohol 
the  mole  fraction  of  the  substance,  water  (molal  weight  = 18),  will 
not  be  identical  with  the  molecular  fraction  of  the  molecular 
species,  H20,  because  there  is  in  the  solution  more  than  one  mo- 
lecular species  corresponding  to  the  substance,  water.  Mole 
fraction  and  molecular  fraction  will  both  be  represented  by  the 
same  symbol,  x,  and  whenever  it  is  necessary  to  distinguish  be- 
tween them  this  will  be  done  in  the  context. 

A common  method  of  expressing  the  composition,  especially  of 
dilute  solutions,  is  in  terms  of  concentration.  By  the  concentra- 
tion, C,  of  any  substance  in  a solution  is  meant  the  quantity  of 
the  substance  per  unit  volume  of  solution.  This  may  be  expressed 
either  as  (1)  moles,  (2)  formula  weights,  or  (3)  equivalent  weights 
of  the  substance  per  liter  of  solution  (I,  7).  The  terms  (1) 
molal,  (2)  formal,  and  (3)  normal,  are  used  correspondingly  to 
indicate  the  strength  of  the  solution.  Thus  a solution  containing 
20.829  grams  (i.e.,  0.1  of  (137.37+2X35.46))  of  barium  chlo- 
ride, BaCl2,  per  liter  is  said  to  be  0.1  molal,  0.1  formal,  and  0.2 
equivalent  or  0.2  normal  with  respect  to  barium  chloride.  Some- 
times 1000  grams  of  solvent  is  made  the  basis  for  expressing  the 
concentration  of  the  solution  instead  of  1000  c.c.  of  solution. 
Concentrations  expressed  on  this  basis  are  termed  weight  concen- 


Sec.  61 


SOLUTIONS  I 


121 


trations  to  distinguish  them  from  the  volume  concentrations 
defined  above.  For  the  same  reason  the  terms  weight-molal, 
weight-formal , and  weight-normal  are  correspondingly  employed. 
Weight  concentrations  can  be  readily  calculated  from  volume 
concentrations  or  vice  versa,  if  the  density  of  the  solution  is  known. 

Problem  3. — A solution  of  density,  D,  is  1/n-weight  formal  with  respect 
to  a substance  whose  formula  weight  is  M.  What  is  its  Volume  concen- 
tration in  formula  weights  per  liter? 

Problem  4. — Calculate  the  weight-formal  concentration  of  the  first- 
named  substance  in  each  of  the  solutions  given  in  problem  2.  The  last- 
named  solution  has  at  20°  a density  of  1:038143  grams  per  cubic  centimeter. 
What  is  its  volume-normal  concentration? 

Problem  5. — Calculate  the  molal  concentration  of  alcohol  (C2H5OH)  in 
a 5 per  cent,  solution  of  it  in  water.  The  density  of  the  solution  at  20°  is 
0.98936.  Calculate  also  its  weight  molal  concentration  and  the  mole  fraction 
of  the  water  (H20)  in  the  solution. 

Problem  6. — A 0.25  formal  aqueous  solution  of  H2SO4  has  at  15°  a den- 
sity of  1.016.  What  per  cent,  of  sulphuric  acid  does  it  contain?  What 
is  its  volume  normality  with  respect  to  sulphuric  acid? 

6.  Vapor  Pressure. — The  vapor  pressure,  p,  of  a solution  is 
equal  to  the  sum  of  the  partial  vapor  pressure  of  its  constituents, 
or 

P = Pa+Pb+  ....  (3) 

7.  Boiling  Point. — The  boiling  point  of  a solution  is  defined  as 
the  temperature  at  which  the  vapor  pressure  of  the  solution  is 
equal  to  the  total  pressure  upon  it.  (Cf.  IV,  3.) 

8.  Freezing  Point.— A solution  has  in  general  as  many  freez- 
ing points  as  there  are  substances  in  the  solution  which  are  able 
to  separate  out  as  pure  crystals.  The  freezing  point  of  a given 
solution  is  the  temperature  at  which  the  solution  is  in  equi- 
librium with  the  pure  crystals  of  one  of  its  constituents.  (Cf. 
VII,  1.)  The  nature  of  the  crystalline  phase  must,  therefore, 
always  be  stated  except  in  the  case  of  dilute  solutions,  where  by 
general  agreement,  unless  otherwise  specified,  the  freezing  point 
is  understood  to  be  the  temperature  at  which  the  solution  is  in 
equilibrium  with  pure  crystals  of  the  constituent  which  is  desig- 
nated as  the  solvent. 


CHAPTER  XII 


SOLUTIONS  II:  THE  COLLIGATIVE  PROPERTIES  OF 
SOLUTIONS  AND  THE  THERMODYNAMIC 
RELATIONS  WHICH  CONNECT  THEM 

1.  Vapor  Pressure  and  Temperature. — We  have  already  noted 
(IV,  3)  that  the  vapor  pressure  of  a substance  always  increases 
with  rise  in  temperature.  In  the  case  of  any  pure  liquid  or  of 
any  constituent  of  a solution  under  a constant  external  pressure, 
P (that  of  the  atmosphere,  for  example),  the  rate  of  increase  of 
the  vapor  pressure  with  rise  in  temperature  is  indicated  mathe- 


subscripts  (in  this  case  P and  x)  indicate  variables  which  are  con- 
stant for  the  process  under  consideration.  The  above  mathemat- 
ical expression  stated  in  words  would  be  read  as  follows:  the 
temperature  rate  of  change  of  the  vapor  pressure  of  a pure  sub- 
stance (or  of  the  partial  vapor  pressure  of  any  constituent  of  a 
solution  in  which  the  mole  fraction  of  the  constituent  is  x)  under 
constant  external  pressure,  P;  or  more  briefly,  the  partial  of  p 
with  respect  to  T,  P and  x constant. 

It  can  be  shown  that  the  Second  Law  of  Thermodynamics 
leads  to  the  following  exact  expression  for  this  differential  coeffi- 
cient: 


(i) 


where  Lv  is  the  molal  heat  of  vaporization  (IV,  2)  of  the  substance 
and  Vo  its  molal  volume  in  the  vapor  state  under  the  conditions 
indicated. 

For  a pure  crystalline  solid  we  have,  similarly, 


(la) 


Ls  being  the  molal  heat  of  sublimation  (VI,  3). 

122 


Sec.  2] 


SOLUTIONS  II 


123 


If  the  substance  is  under  its  own  vapor  pressure  at  all  temper- 
atures, instead  of  under  a constant  external  pressure,  equations 
(1)  and  (la)  have  the  form 


'dp\  = 

<)T i x,p  = p (vq  — Vq)T 


(2) 


where  Fo  is  the  molal  volume  of  the  liquid  (or  crystals)  at  the 
pressure  P and  temperature  T.  This  is  known  as  the  Clausius  a- 
Clapeyron6  equation.  Except  in  the  neighborhood  of  the  critical 
point,  Fo  is  usually  so  small  in  comparison  with  v0  that  it  may  be 
neglected  and  then  equation  (2)  takes  the  form  of  equation  (1), 
that  is,  equation  (1),  which  is  rigorously  exact  if  P is  constant, 
is  also  approximately  correct  for  many  cases  where  P = p. 


Problem  1. — In  order  to  integrate  equation  (1)  or  (la)  it  is  first  necessary 
to  express  L and  Vo  as  functions  of  one  of  the  variables.  If  the  vapor  obeys 
the  perfect  gas  law  and  if  L is  a constant  with  respect  to  variations  in  T, 
show  that  the  integral  of  the  above  equations  has  the  form, 

1 pi  L / 1 1 \ 

logen~R\T^~T'i)  (3) 

Problem  2. — At  —2°  water  has  a vapor  pressure  of  3.952  mm.  of  Hg. 
Its  heat  of  vaporization  at  0°  is  2495  joules  per  gram.  Calculate  its  vapor 
pressure  at  2°. 


2.  Variation  of  Boiling  Point  with  External  Pressure  on  the 
Liquid. — Since  by  definition  (XI,  7)  vapor  pressure,  p,  and  exter- 
nal pressure,  P,  for  any  liquid  are  equal  to  each  other  at  the  boil- 
ing point,  Tb,  of  the  liquid,  it  follows  that  at  this  temperature 


c )Tb 
dP 


l c )P~ 


or 


i /m a 

Tb\c)P  L Lv 


■(£). 

(4) 

we  find 

(^o  — Fo  )T  B 

Lv 

(5) 

= ir-  (approx.) 

Liv 

(6) 

“ Rudolph  Julius  Emmanuel  Clausius  (1828-1888).  Professor  of  Physics 
in  the  University  of  Zurich,  the  University  of  Wiirtzburg  and  until  his  death 
in  the  University  of  Bbnn.  One  of  the  founders  of  modern  thermodynamics. 

6 Benoit-Paul  Emile  Clapeyron,  (1799-1864).  French  engineer  and  pro- 
fessor at  L’Ecole  de  Ponts  et  Chaussees. 

9 


124 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XII 


Problem  3. — Compare  the  purely  thermodynamic  relationship  expressed 

1 dT 

by  equation  (6)  with  the  Crafts’  equation,  — — ,„  = const.  (IV,  5).  What 

■l  Bo  Oi 

Lv 

can  you  state  with  regard  to  the  ratio,  — for  pure  liquids  at  their  boiling 

Vq 

points?  Compare  also  with  Trouton’s  rule,  and  draw  a conclusion  with 

regard  to  the  ratio  for  related  liquids. 

1 J3o 


3.  Effect  of  Pressure  upon  Vapor  Pressure. — If  the  pressure 
upon  any  pure  substance  in  the  liquid  (or  crystalline)  state  be 
increased,  the  vapor  pressure  of  the  substance  also  increases,  the 
quantitative  thermodynamic  relation  between  the  two  being 
expressed  by  the  equation, 


'Z >p\  _ Vo 

pP/T 


(7). 


where  Vo  is  the  molal  volume  of  the  substance  in  the  liquid  (or 
crystalline)  state  at  the  pressure,  P,  and  temperature,  T.  An 
equation  of  the  same  form, 


pPA\  = Voa 
Z)P  )t,x  Voa 


also  holds  for  the  effect  of  pressure  upon  the  partial  vapor  pres- 
sure, pA,  of  any  constituent  of  a solution.  Voa?  the  partial  molal 
volume  of  A is  the  mixture,  is  equal  to  the  increase  in  the  volume 
of  an  infinite  amount  of  the  solution,  which  takes  place  when  one 
mole  of  the  substance,  A,  is  added  to  it. 

4.  Vapor  Pressure  Lowering. — If  to  any  pure  liquid,  A,  having 
the  vapor  pressure,  p0,  we  add  some  substance  which  forms  a 
solution  with  it,  it  can  be  shown  both  from  moleculaf  kinetics 
and  from  the  Second  Law  of  Thermodynamics  that  the  vapor 
pressure  of  A is  thereby  lowered.  In  other  words,  the  partial 
vapor  pressure  of  any  substance  from  a solution  is  always  lower 
than  its  vapor  pressure  in  the  pure  state.  The  vapor  pressure 
lowering,  A p,  is  defined  by  the  equation, 

Ap  = p0-p  (9) 

and  a quantity  called  the  relative  vapor  pressure  lowering  is 

defined  as  --  ^ or  where  po  is  the  vapor  pressure  0f  the  sub- 

Po  Po 

stance  as  a pure  liquid  and  p its  partial  vapor  pressure  from  the 


Sec.  6] 


SOLUTIONS  II 


125 


solution,  both  being  at  the  same  temperature,  T,  and  under  the 
same  external  pressure,  P. 

5.  Elevation  of  the  Boiling  Point. — Problem  4— From  what  has 
just  been  said  regarding  the  lowering  of  vapor  pressure  which  occurs 
when  one  substance  is  dissolved  in  another,  show  that  the  following  state- 
ment must  also  be  true:  The  boiling  point  of  a liquid  is  always  raised  by 
dissolving  in  it  any  substance  whose  own  vapor  pressure  is  negligibly  small. 


The  elevation  of  the  boiling  point,  or  the  boiling  point  raising 

as  it  is  also  called,  is  defined  by  the  equation, 

A TB  = TB-TBo  (10) 

where  TB  is  the  boiling  point  of  the  solution  and  TBo  that  of  the 
pure  liquid,  both  boiling  points  being  of  course  for  the  same  pres- 
sure. If  the  other  constituent  of  the  solution  has  an  appreciable 
vapor  pressure  of  its  own,  • the  boiling  point  is  not  necessarily 
raised  when  the  solution  is  formed  but  may  even  be  lowered,  if 
the  other  constituent  is  a very  volatile  substance.  This  occurs, 
for  example,  when  ether  is  dissolved  in  alcohol. 

6.  Freezing  Point  Lowering. — We  have  already  seen  (X,  10) 
that  at  the  freezing  point  of  a pure  liquid  the  liquid  has  the  same 
vapor  pressure  as  the  crystals  with  which  it  is  in  equilibrium. 
If  to  such  a system  a second  substance  is  added  which  dissolves 
in  the  liquid,  but  not  in  the  crystals,  the  vapor  pressure  of  the 
liquid  is  thereby  lowered  and  it  will  no  longer  be  in  equilibrium 
with  the  crystalline  phase.  The  crystals,  since  they  now  have 
the  higher  vapor  pressure,  will  tend  to  pass  over  into  the  solution 
which,  if  the  two  are  in  contact,  they  can  do  simply  by  melting. 
The  process  of  melting  is,  however,  attended  by  an  absorption  of 
heat  (VII,  2),  and  consequently  the  whole  system  will  cool  down 
until  some  temperature  is  reached  where  the  vapor  pressure  of 
the  substance  in  the  crystalline  state  is  again  the  same  as  its 
partial  vapor  pressure  from  the  solution  and  hence  the  two  phases 
become  once  more  in  equilibrium  with  each  other.  That  such 
a temperature  will  be  reached  can  be  demonstrated  thermody- 
namically and  can  also  be  seen  from  a study  of  Fig.  16.  This 
temperature  we  have  already  defined  (XI,  8)  as  the  freezing  point 
of  the  solution  and  the  freezing  point  of  a liquid  is,  therefore 


126 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XII 


evidently  always  lowered  by  dissolving  another  substance  in  it. 
The  freezing  point  lowering  is  defined  by  the  equation, 

A Tf=Tf-Tf  (11) 

where  TFo  is  the  freezing  point  of  the  pure  liquid  and  TF  that  of 
the  solution. 


-22°  -17°  -12°  7°  0°  3°  8°  13°  18°  23°  28° 

.Fig.  16. 


7.  Osmotic  Pressure  and  Osmosis. — Consider  two  vessels 
placed  side  by  side  under  a bell-jar  as  shown  in  Fig.  17.  Let 
vessel  number  1 contain  any  solution,  made  up  of  any  number 
of  constituents,  A,  B,  C,  etc.,  and  let  vessel  number  2 contain 
any  one  of  these  constituents,  A for  example,  in  the  form  of  a 
pure  liquid.1  The  partial  vapor  pressure  of  A from  the  first 
vessel  is  less  than  its  vapor  pressure  from  the  second  vessel  (XII,  4) 
and  hence  there  will  be  a tendency  for  the  substance,  A,  to  pass 
over  from  the  pure  liquid  state  (vessel  2)  into  the  solution  (vessel 

1 In  the  most  general  case  vessel  number  2 might  contain  constituent  A, 
dissolved  in  some  other  pure  liquid  to  form  a solution  of  any  stipulated 
strength.  This  other  pure  liquid  would,  in  such  a case,  serve  simply  as  a 
standard  reference  liquid  and  for  theoretical  purposes  might  be  some  wholly 
hypothetical  liquid,  endowed  with  any  desired  properties.  Unless  a standard 
reference  liquid  of  some  kind  is  specified,  however,  the  constituent  in  question 
is  usually  considered  as  present  in  the  pure  liquid  state  in  the  second  vessel. 


Sec.  7l 


SOLUTIONS  II 


127 


1)  which  it  might  do  by  distillation,  for  example.  Now  as  ex- 
plained above  (XII,  1)  the  vapor  pressure  of  A from  the  two  vessels 
could  be  made  the  same  either  by  raising  the  temperature  of  the 
solution  or  by  lowering  that  of  the  pure  liquid,  the  total  pressure 
on  both  remaining  the  same.  Or,  the  two  vapor  pressures  might 
also  be  made  the  same  by  keeping  the  temperature  of  both 
vessels  constant  but  varying  the  total  pressure  on  one  of  them, 
i.e.}  by  increasing  the  total  pressure  on  the  solution  or  by  decreas- 
ing the  total  pressure  on  the  pure  liquid  (XII,  3). 


The  difference  in  the  total  pressure  upon  the  two  vessels 
which  is  just  sufficient  to  produce  equality  in  the  vapor  pressures 
of  A from  both  vessels  is  called  the  osmotic  pressure  of  the  solu- 
tion with  reference  to  constituent  A and  will  be  indicated  by  the 
symbol,  nA.  In  order  to  completely  define  this  pressure  differ- 
ence, the  actual  pressures  on  the  two  vessels  must  also  be  speci- 
fied. Unless  otherwise  stated,  therefore,  we  shall  define  the 
osmotic  pressure,  nA,  by  the  equation, 

nA=P-PA  (12) 

where  PA  is  the  pressure  on  the  pure  liquid,  A,  and  P that  on  the 


128 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XII 


solution,  when  A has  the  same  vapor  pressure 2 from  both,  and 
unless  otherwise  specified  P will  always  be  understood  to  be  one 
atmosphere. 

Problem  6. — Show  by  a method  of  reasoning  similar  to  that  employed  in 
X,  10,  that  if  a pure  liquid,  A,  be  separated  from  a solution  containing  it 
by  a membrane  which  is  'permeable  only  to  molecules  of  A,  then  A will  pass 
through  the  membrane  into  the  solution,  if  the  solution  and  the  pure  liquid 
are  both  at  the  same  temperature  and  pressure;  but  that  if  the  pressure 
difference  IIa  (as  just  defined)  be  established  on  the  two  liquids,  then  there 
will  no  longer  be  any  tendency  for  A to  pass  through  the  membrane  into  the 
solution. 

The  passage  of  a liquid  through  such  a membrane  (called  a 
semipermeable  membrane)  is  termed  osmosis.  Semipermeable 
membranes  are  found  in  all  animal  and  vegetable  organisms  and 
osmosis  plays  a very  important  role  in  physiological  processes. 
The  molecular  kinetic  interpretation  of  the  process  of  osmosis 
will  be  discussed  later.  The  phenomenon  of  osmosis  can  be 
easily  demonstrated  as  a lecture  experiment  by  tying  a piece  of 
gold  beater’s  skin  over  the  mouth  of  a thistle  tube,  filling  the 
tube  with  a strong  solution  of  sugar  colored  with  a little  cochi- 
neal and  then  immersing  the  inverted  tube  in  a beaker  of  water, 
as  shown  in  Fig.  18.  The  gold  beater’s  skin  acts  as  a semi- 
permeable membrane  permitting  water  to  pass  through  it  readily 
but  not  sugar.  Water,  therefore,  passes  through  the  membrane 
into  the  sugar  solution  and  dilutes  it,  causing  the  volume  to 
increase  and  the  level  of  the  solution  in  the  tube  to  rise.  The 
weight  of  the  column  of  solution  above  the  membrane  and  hence, 

2 More  generally  stated  the  definition  of  IIa  would  be  IIa  =P  — Pa,  where 
Pa  is  the  pressure  on  the  pure  liquid,  A,  and  P that  on  the  solution  when  A 
has  the  same  escaping  tendency  from  both.  The  vapor  pressure  of  the  mole- 
cules of  the  substance,  A,  from  any  phase  or  system  containing  it  is  only  one 
of  the  many  ways  in  which  this  escaping  tendency  may  manifest  itself. 
It  is  one  which  is  very  readily  visualized  by  the  student,  however,  and  for 
that  reason  we  shall  employ  it  frequently.  Increase  of  pressure  upon  a 
liquid  not  only  increases  its  vapor  pressure,  that  is,  its  tendency  to  escape 
into  the  vapor  phase,  but  it  also  increases  its  tendency  to  escape  into  any 
other  condition  whatsoever,  as  can  be  readily  shown  by  purely  thermody- 
namic reasoning.  While,  therefore,  the  concept  of  escaping  tendency  in 
general  is  a more  abstract  method  of  expression,  we  shall  for  the  present 
employ  instead  the  more  concrete  concept  of  its  manifestation  as  a vapor 
pressure. 


Sec.  7] 


SOLUTIONS  II 


129 


therefore,  the  pressure  upon  the  layer  of  solution  next  the  mem- 
brane gradually  increases  and  with  a properly  prepared  mem- 
brane osmosis  will  continue  until  the  resulting  pressure  becomes 
so  great  that  the  tendency  of  the  water  to  pass  through  the 
membrane  into  the  solution  is  just  equal  to  its  tendency  to 
pass  in  the  opposite  direction,  and  consequently  the  rates  of 
osmosis  in  the  two  directions  balance  each  other 
and  the  column  of  liquid  in  the  thistle  tube 
ceases  to  rise.  When  this  condition  is  reached 
the  weight  of  liquid  above  the  semipermeable 
membrane  is  a measure  of  the  osmotic  pressure 
of  the  sugar  solution,  with  reference  to  the  con- 
stituent, water. 

This  particular  osmotic  pressure,  n'A,  how- 
ever, is  obviously  defined  by  the  equation, 

U'=P-PA  = (P-1)  atmos.  (13) 

where  P is  the  pressure  upon  the  layer  of  solu- 
tion next  to  the  semipermeable  membrane.  It 
is  the  osmotic  pressure  as  defined  by  equation 
(13)  rather  than  that  defined  by  equation  (12) 
which  is  usually  obtained  in  direct  osmotic  pres- 
sure measurements.  That  is,  it  is  the  pressure 
difference  necessary  to  establish  equilibrium 
when  the  pure  liquid  (instead  of  the  solution) 
is  under  atmospheric  pressure.  Either  osmotic 
pressure  nA  or  n'A  may,  however,  be  thermodynamically  calcu- 
lated, if  the  other  is  known,  and  in  the  case  of  aqueous  sugar  solu- 
tions of  moderate  concentrations  the  two  osmotic  pressures  are 
practically  identical.  The  one  defined  by  equation  (12)  is, ' 
however,  in  general  a simpler  one  to  employ  than  is  the  one 
defined  by  equation  (13). 

In  direct  osmotic  pressure  measurements  with  aqueous  solu- 
tions the  membrane  employed  is  usually  a film  of  cupric  ferro- 
cyanide  deposited  in  the  pores  of  a porous  earthern  cup.  Direct 
measurements  of  osmotic  pressure  are  difficult  to  carry  out,  how- 
ever, and  have  thus  far  been  accurately  made  only  in  the  case  of 
a few  aqueous  solutions.  The  osmotic  pressure  of  any  solution 
can  be  calculated  thermodynamically,  however,  from  the  freez- 


130 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XII 


ing  point  or  from  the  vapor  pressure  of  the  solution  and  this  is, 
in  the  majority  of  cases,  the  most  reliable  as  well  as  the  most  con- 
venient method  to  employ  in  case  one  desires  to  know  the  osmotic 
pressure  of  any  solution.  The  role  played  by  osmotic  pressure 
in  the  theory  of  solutions  is  of  such  a nature,  however,  that  the 
knowledge  of  the  numerical  value  of  the  osmotic  pressure  of  any 
solution  is  seldom  of  much  importance.  Osmotic  pressure  is 
chiefly  of  value  simply  as  a concept  by  means  of  which  some  of 
the  processes  employed  in  the  derivations  of  the  laws  of  solution 
may  be  conveniently  visualized.  Historically,  however,  it  has 
played  and  to  many  chemists  of  the  present  day  still  plays  an 
important  role  in  solution  theory,  chiefly  owing  to  some  popular 
misconceptions  as  to  its  nature  and  the  analogy  between  it  and 
gas  pressure,  as  will  be  explained  further  in  Chapter  XIV.  For 
this  reason  more  space  will  be  given  to  the  discussion  of  osmotic 
pressure  than  would  be  justified  by  its  actual  importance  in  the 
theory  of  solutions. 

8.  The  Thermodynamic  Relations  Connecting  the  Colligative 
Properties  of  a Solution. — The  magnitudes  of  the  vapor  pressure 
lowering,  the  boiling  point  raising,  the  freezing  point  lowering, 
the  osmotic  pressure,  and  certain  other  allied  properties  of  a 
solution  depend  in  general  upon  the  molal  composition  of  the  solu- 
tion and  upon  the  natures  of  its  constituents.  But  in  the  case  of 
an  important  class  of  solutions  which  will  be  discussed  in  the  next 
chapter,  the  magnitude  of  these  quantities  for  one  constituent 
of  the  solution  depends  only  upon  the  nature  of  this  constituent 
and  upon  its  mole  fraction  in  the  solution  and  not  at  all  upon  the 
natures  of  the  other  components  of  the  solution  nor  the  relative 
amounts  of  them  present. 

Now  although,  in  general,  the  magnitude  of  any  one  of  these 
quantities  depends  both  upon  the  composition  of  the  solution  and 
the  natures  of  its  constituents,  the  relation  between  any  two  prop- 
erties for  any  constituent  of  any  given  solution  depends  only 
upon  the  nature  of  this  constituent  and  not  at  all  upon  the  mole 
fraction  of  the  constituent  in  the  solution  nor  upon  the  number 
or  natures  of  the  other  constituents  of  the  solution.  Thus,  for 
example,  the  relation  connecting  the  partial  vapor  pressure  of 
any  constituent,  A,  of  a solution  with  its  osmotic  pressure  does 
not  depend  in  any  way  upon  the  per  cent,  of  A in  the  solution 


Sec.  8] 


SOLUTIONS  II 


131 


4 


nor  upon  the  number,  amounts,  or  natures  of  the  other  substances 
in  the  solution.  For  these  reasons  the  properties  mentioned  are 
called  the  colligative  properties  of  the  solution. 

The  exact  and  general  relations  which  connect  the  colligative 
properties  of  a solution  with  one  another  can  be  easily  derived 3 
from  the  First  and  Second  Laws  of  Thermodynamics  without  any 
additional  assumptions.  Relations  of  this  character,  that  is, 
relations  which  are  necessary  consequences  of  the  two  laws  of 
energy  alone,  will  be  referred  to  as  “ purely  thermodynamic 
relations.”  We  shall  not  stop  here  to  explain  further  the  details 
of  the  derivations  of  these  relations  but  will  simply  state  a few 
of  them,  using  the  nomenclature  already  employed. 

(a)  Vapor  pressure,  p,  and  freezing  point,  TF,  both  with  ref- 
erence to  the  same  constituent,  A: 


'*Pa\  Ls 

pTFj  p,  xA  v0Tf 


(14) 


where  Ls  is  the  molal  heat  of  sublimation  (VI,  6)  of  the  pure  con- 
stituent at  the  pressure  P,  and  the  temperature  T„  and  ^ois  the 
molal  volume  of  its  vapor  at  the  pressure  pA,  and  the  tem- 
perature, Tf. 

(b)  Vapor  pressure  and  osmotic  pressure  for  constituent,  A. 


pVA\  = _Vj 

dIIA/  T,  XA  Vo 


(15) 


Vo  being  the  molal  volume  of  A in  the  liquid  state  at  the  tempera- 
ture T,  and  the  pressure  1— nA,  and  Vo  the  molal  volume  of  the 
vapor  at  the  pressure  p,  and  the  temperature  T. 

(c)  Vapor  pressure  and  boiling  point,  TB : Since  in  this  case  by 
definition  (XI,  7)  p=P,  the  relation  is  simply  equation  (5)  above, 
which  may  be  written: 

(dTs)  x=Jv7^VojlrB  (16) 

(d)  Osmotic  pressure  and  freezing  point,  both  with  reference 
to  the  same  constituent,  A: 

/dnA\  —Lf 
\dTF)  “Foa  Tf 

3 See  Jour.  Amer.  Chem.  Soc.,  32,  496  and  1636  (1910). 


(17) 


132 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XII 


To  these  relations  might  be  added  many  others,  but  the  state- 
ment of  additional  ones  will  be  deferred  until  they  are  needed. 
It  will  be  noticed  that  none  of  the  above  equations  contains  any 
quantity  which  is  in  any  way  dependent  upon  the  amount  of  the 
substance,  A,  present  in  the  solution  or  upon  the  number,  nature, 
or  amounts  of  the  other  constituents  of  the  solution. 

There  is,  however,  for  every  solution,  a set  of  relations  which 
connects  each  of  the  colligative  properties  of  the  solution  with 
the  nature  of  its  constituents  and  its  molal  composition.  This 
set  of  relations  we  shall  call  the  “Laws  of  the  Solution.”  It  is 
evident,  from  what  has  just  been  said,  that  in  order  to  deduce 
the  complete  set  of  these  laws  for  any  given  solution,  it  is  neces- 
sary to  have  only  one  of  the  laws  as  a starting  point,  for  all  of  the 
others  can  then  be  obtained  by  combining  this  one  law  with  the 
purely  thermodynamic  relations  discussed  above.  As  our 
starting  point  in  deducing  the  Laws  of  Solutions  we  shall  employ 
the  law  connecting  the  partial  vapor  pressure  of  any  constituent 
of  a solution  with  its  mole  fraction  in  the  solution. 


Problem  1. — If  the  vapor  is  a perfect  gas  and  To  is  independent  of  7 r, 
show  that  the  integral  of  equation  (15)  is 


II 


RT 
V 0 


logc 


P 

Po 


(18) 


where  po  is  the  vapor  pressure  of  pure  liquid  A at  the  temperature  T. 
also  that  this  integral  may  be  written  in  the  form, 

See  Ref.  3 for  the  integrations  of  equations  (14)  to  (17). 


Show 


(19) 


9.  The  Fundamental  Vapor  Pressure  Law. — If  in  a solution 
having  the  two  molecular  components,  A and  B,  the  molecular 
fraction  of  A be  increased  from  xA  to  Za  + cIzaj  the  corresponding 
increase,  dpA>  in  its  partial  vapor  pressure,  pA,  is  given  by  the 
expression, 

dpA=iA(T.E.)dxA  (20) 

and  similarly  for  component  B, 

dpB=iB(T.E.)dxB  (21) 

the  temperature  and  total  pressure  being  constant;  or  stated  in 
words,  when  the  molecular  fraction  of  any  component  of  a solu- 


Sec.  9] 


SOLUTIONS  II 


133 


tion  is  increased  by  a very  small  amount,  dz,  the  corresponding 
increase,  dp,  in  its  partial  vapor  pressure  above  the  solution  is 
equal  to  dx  multiplied  by  a quantity  which  is  a function  of  the 
thermodynamic  environment  (see  XIII,  1)  which  prevails  in 
the  interior  of  the  solution.  It  is  evident  that  in  order  to  inte- 
grate equation  (20)  it  is  first  necessary  to  know  the  form  of  the 
function,  f A(T.E.).  The  form  of  this  function  and  the  values  of 
its  parameters  will  in  general  be  determined  by  the  number, 
nature  and  relative  proportions  of  the  different  constituents  of 
the  solution,  and  since  in  general  these  may  be  of  the  most  va- 
ried character  it  is  not  practicable  to  attempt  an  evaluation 
of  fA  ( T.E .)  which  will  hold  for  all  possible  types  of  solutions. 
A better  procedure  is  to  classify  solutions,  as  far  as  possible, 
with  reference  to  this  point,  and  then  to  seek  an  evaluation  of 
fA  {T.E.)  for  each  class. 

As  our  knowledge  of  solutions  has  progressed  we  have  come  to 
recognize  two  large  groups  or  classes  of  solutions  for  which  it  is 
possible  to  evaluate  this  function  in  a satisfactory  manner,  or  in 
other  words,  for  which  it  is  possible  to  construct  more  or  less 
complete  and  satisfactory  systems  of  laws  and  theories.  These 
two  classes  may  be  called,  respectively,  (1)  Solutions  of  Constant 
Thermodynamic  Environment  and  (2)  Solutions  whose  Thermo- 
dynamic Environment  is  a Function  of  the  Ion-concentration. 
We  shall  restrict  our  consideration  of  the  Theories  of  Solution  to 
these  two  classes  and  in  building  up  the  system  of  laws  for  each 
class  we  shall  follow  the  logical  rather  than  the  historical  method 
of  development,  as  our  main  purpose  will  be  to  present  as  clear 
and  complete  a picture  as  possible  of  the  present  condition  of 
our  systematized  knowledge  of  solutions  rather  than  to  trace 
the  stages  by  which  this  condition  has  been  reached. 


CHAPTER  XIII 


SOLUTIONS  III:  THERMODYNAMIC  ENVIRONMENT. 

IDEAL  SOLUTIONS  AND  DILUTE  SOLUTIONS 

1.  Thermodynamic  Environment. — Consider  the  two  liquids, 
mercury,  Hg,  and  benzene,  C6H6,  two  substances  which  in  their 
chemical  and  physical  properties  are  widely  different  from  each 
other.  If  we  shake  these  two  liquids  together  in  a test-tube  we 
find  that  neither  substance  will  dissolve  in  the  other  to  an  appre- 
ciable extent.  Instead,  we  obtain  two  liquid  layers,  one  of  which 
is  practically  pure  mercury  and  the  other  practically  pure  ben- 
zene. The  two  species  of  molecules,  Hg  and  C6H6,  have  scarcely 
any  attraction  for  each  other  and  refuse  to  intermingle.  In 
other  words,  the  substance  mercury  is  not  capable  of  existing 
in  a molecularly  dispersed  state  (XI,  1)  under  the  conditions 
which  prevail  in  liquid  benzene,  nor,  on  the  other  hand,  are 
benzene  molecules  able  to  exist  in  the  molecularly  dispersed  state 
in  surroundings  containing  mercury  molecules.  This  inability 
of  these  two  liquids  to  mix  with  each  other  is  closely  connected 
with  the  great  chemical  and  physical  differences  between  them. 

Suppose  now  that  we  replace  the  mercury  by  water  [H20 
+ (H20)2  + (H20)3-b  . . . ],  a liquid  whose  molecules  resem- 
ble those  of  benzene  more  closely  than  mercury  molecules  do, 
and  consequently  (I,  2)  one  which  in  its  physical  and  chemical 
properties  is  more  like  benzene  than  mercury  is.  On  shaking 
water  and  benzene  together  we  obtain,  just  as  before,  two  liquid 
layers,  a water  layer  below  and  a benzene  layer  above.  On  care- 
ful examination,  however,  we  would  find  that  the  water  layer 
contains  a little  benzene  (about  0.1  per  cent.)  dissolved  in  it  and 
that  the  benzene  layer  contains  a little  water  (about  0.03  per  cent.) 
dissolved  in  it.  These  two  liquids  are  able,  therefore,  to  mix 
with  each  other  to  a very  slight  but  quite  appreciable  extent. 
In  other  words,  an  appreciable  quantity  of  water  is  capable  of 

134 


Sec.  1] 


SOLUTIONS  III 


135 


existing  in  the  molecularly  dispersed  state  in  an  environment 
which  is  made  up  almost  entirely  of  benzene  molecules  and  simi- 
larly an  appreciable  quantity  of  benzene  is  able  to  exist  in  the 
molecularly  dispersed  condition  in  an  environment  made  up 
almost  entirely  of  water  molecules. 

Let  us  now  take  a third  liquid,  ethyl  alcohol,  which  resembles 
benzene  even  more  closely  than  does  water.  If  we  shake  benzene, 
C6H6,  and  alcohol,  C2H60,  together  we  find  that  we  always  obtain 
a homogeneous  system  whatever  be  the  relative  amounts  of 
benzene  and  of  alcohol  taken,  that  is,  these  two  liquids  are 
miscible  with  each  other  in  all  proportions.  The  attractive 
forces  of  alcohol  molecule  for  alcohol  molecule,  benzene  molecule 
for  benzene  molecule,  and  alcohol  molecule  for  benzene  molecule 
are  so  related  that  when  we  pour  alcohol  into  benzene  the  alcohol 
molecules  intermingle  with  those  of  benzene  and  continue  to  do 
so  in  whatever  numbers  they  are  added.  The  first  molecules  of 
alcohol  which  enter  the  benzene  find  themselves  surrounded  by 
benzene  molecules  only.  As  the  amount  of  alcohol  is  increased 
the  nature  of  the  medium  surrounding  any  given  alcohol  or  benzene 
molecule  changes  gradually  from  one  composed  almost  entirely 
of  benzene  molecules  to  one  in  which  the  proportion  of  alcohol 
molecules  gradually  increases  until  finally  the  medium  surround- 
ing any  given  alcohol  molecule  or  benzene  molecule  is  composed 
almost  entirely  of  alcohol  molecules. 

Now  the  tendency  of  a given  molecule  to  escape  from  a solution 
containing  it  depends  upon  the  conditions  which  prevail  within 
that  particular  solution.  The  molecule  is  subject  to  the  action 
of  various  attractive  and  repulsive  forces  as  well  as  to  collisions 
from  the  molecules  which  surround  it  and  the  sum  total  of  all 
these  environmental  influences  determines  the  magnitude  of 
the  escaping  tendency  of  the  molecule  in  question.  An  attempt 
to  analyze  further  the  nature  of  these  environmental  influences 
would  in  the  present  state  of  our  knowledge  be  largely  speculative 
and  would  have  no  particular  value.  It  is,  however,  important 
to  recognize  the  existence  of  these  influences  and  their  general 
character,  and  it  will  be  convenient  to  have  a name  to  designate 
the  sum  total  of  these  effects.  The  nature  of  the  medium  sur- 
rounding any  given  molecular  species  in  a solution  will,  therefore, 
be  called  the  thermodynamic  environment  of  this  molecular 


136 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIII 


species.  The  thermodynamic  environment  which  prevails  within 
a solution  depends,  in  general,  upon  the  relative  numbers  and  the 
kinds  of  molecules  which  make  up  the  solution  and  upon  the  tem- 
perature and  the  pressure.1 

When  the  two  molecular  species  which  make  up  the  solution 
are  very  different  in  character  the  thermodynamic  environment 
prevailing  within  the  solution  will  be  so  different  from  that  which 
prevails  within  one  of  the  two  pure  liquids,  that  a separation  into 
two  liquid  layers  will  occur  when  the  proportions  of  the  two  con- 
stituents reach  certain  values  which  are  determined  by  the  nature 
of  the  constituents  and  by  the  temperature  and  the  pressure. 
Thus  when  we  add  water  to  benzene  the  water  molecules  inter- 
mingle at  first  with  those  of  the  benzene,  forming  a solution  which 
has  a thermodynamic  environment  practically  the  same  as  that 
which  prevails  in  pure  benzene.  When  the  concentration  of  the 
water  molecules  reaches  a certain  value,  however  (which  depends 
upon  the  temperature  and  the  pressure),  any  further  molecules  of 
water  added  will  not  go  into  solution,  since  for  concentrations 
higher  than  this  value  the  mutual  attractions  of  water  molecule 
for  water  molecule  become  so  great  that  any  excess  of  water 
separates  out  as  a new  liquid  layer,  having  a decidedly  different 
thermodynamic  environment. 

In  the  case  of  alcohol  and  benzene  which  mix  with  each  other 
in  all  proportions  the  two  molecular  species  display  many 
differences  in  both  physical  and  chemical  properties  and  the 
thermodynamic  environments  in  the  two  pure  liquids  are  prob- 
ably quite  appreciably  different  from  each  other,  so  that  when 
alcohol  is  poured  into  benzene,  the  thermodynamic  environment 
in  the  solution  changes  gradually  from  that  which  prevails  in 
pure  benzene  to  that  which  prevails  in  pure  alcohol,  but  the  total 
change  in  this  instance  is  not  so  great  but  that  both  species  of 
molecules  are  able  to  adapt  themselves  to  it  and  hence  do  not 
find  it  necessary  to  form  two  distinct  liquid  layers  possessing 
different  thermodynamic  environments. 

To  recapitulate,  therefore,  if  molecules  of  a liquid,  A,  be  in- 
troduced into  a pure  liquid,  B,  they  intermingle  with  those  of 

1 And  in  some  special  cases,  which  will  not  be  considered  here,  upon  the 
amount  and  kind  of  light  with  which  the  solution  is  illuminated  and  upon 
the  magnetic  and  electric  condition  of  its  surroundings. 


Sec.  1] 


SOLUTIONS  III 


137 


B,  forming  a solution.  As  the  concentration  of  the  A molecules 
increases,  two  tendencies  become  manifest.  First,  the  mutual 
attractions  among  the  A molecules  increase  owing  to  the  fact 
that  these  molecules  are  getting  closer  together  as  their  con- 
centration increases  and  as  a result  of  this  greater  attraction  there 
is  a tendency  for  these  molecules  to  collect  together  and  to  form 
themselves  (together  with  some  B molecules)  into  a second  liquid 
layer  having  a different  thermodynamic  environment  from  the 
solution. 

Opposed  to  this  tendency  toward  separation  and  formation  of 
a new  thermodynamic  environment  is  the  fact  that  the  increasing 
numbers  of  A molecules  by  their  very  presence  in  the  solution 
are  changing  the  thermodynamic  environment  of  the  solution 
and  are  making  it  more  like  that  which  would  exist  in  the  layer 
which  tends  to  be  formed  by  the  mutual  attractions  of  the  A 
molecules.  This  naturally  results  in  lessening  the  tendency  of 
the  A molecules  to  separate  and  form  a second  phase.  When 
the  two  molecular  species  are  greatly  different  from  each  other 
the  first  of  these  tendencies  is  likely  to  prove  the  stronger  and  two 
liquid  layers  are  formed.  The  more  nearly  the  two  species  of 
molecules  resemble  each  other  the  greater  will  be  the  second 
tendency  and  when  they  are  sufficiently  alike  the  second  tend- 
ency will  predominate  and  the  two  substances  will  mix  in  all 
proportions. 

Unless  the  two  molecular  species  resemble  each  other  very 
closely,  however,  the  process  of  mixing  is  accompanied  by  quite 
an  appreciable  variation  in  thermodynamic  environment  in  pass- 
ing from  one  pure  liquid  through  the  series  of  mixtures  to  the 
other  pure  liquid,  and  although  the  two  molecular  species  may  be 
able  to  adapt  themselves  to  the  new  thermodynamic  environment 
and  remain  in  solution  together,  this  adaptation  will  in  general  be 
accompanied  by  external  evidences  in  the  way  of  heat  effects 
(evolution  or  absorption  of  heat)  and  volume  changes  (expansion 
or  contraction)  which  take  place  when  the  two  liquids  are  mixed 
together.  The  magnitudes  of  these  effects  may  be  taken  as  a 
rough  indication  of  the  extent  of  the  change  in  thermodynamic 
environment,  in  all  cases  where  the  process  of  mixing  consists 
simply  in  the  intermingling  of  the  two  molecular  species  and  is 
not  accompanied  by  chemical  reactions.  Thus  when  one  mole 


138 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIII 


of  benzene  is  poured  into  a very  large  quantity  of  alcohol  the 
change  in  thermodynamic  environment  undergone  by  the  benzene 
molecules  is  accompanied  by  an  absorption  of  0.36  calorie  of 
heat  and  by  an  appreciable  change  in  volume,  which  indicates  an 
appreciable  but  not  very  large  change  in  thermodynamic  environ- 
ment. When  one  mole  of  alcohol  is  poured  into  a large  quantity 
of  benzene  the  process  is  accompanied  by  an  absorption  of  4 
calories  of  heat.  In  this  case,  however,  we  have  a chemical 
reaction,  namely,  (C2H50H)X  = xC2H5OH,  accompanying  the 
process  of  mixing,  for  ethyl  alcohol  is  an  associated  liquid  (III, 
5)  and  when  it  is  poured  into  a sufficiently  large  quantity  of  ben- 
zene the  associated  molecules  must  (as  we  shall  learn  later) 
dissociate  into  simple  ones  and  this  reaction  will  be  attended  by 
a heat  effect  which  will  be  added  to  that  due  to  the  change  in 
thermodynamic  environment  alone.  The  total  heat  effect  in 
such  a case  will  not,  therefore,  be  a trustworthy  indication  of  the 
extent  of  the  change  in  thermodynamic  environment. 

A better  measure  of  the  magnitude  of  this  change  is  furnished 
by  the  decrease  in  free  energy  (X,  9)  which  accompanies  the 
transfer  of  one  mole  of  one  substance  from  one  thermodynamic 
environment  to  the  other.  Thus,  when  one  mole  of  benzene  is 
transferred  from  the  thermodynamic  environment  which  prevails 
in  pure  benzene  to  that  which  prevails  in  pure  alcohol,  the  change 
in  thermodynamic  environment  will  be  measured  by  the  mag- 
nitude of  the  corresponding  free  energy  decrease,  which  can  be 
shown  to  be  substantially  that  given  by  the  expression, 


(i) 


where  p o is  the  vapor  pressure  of  the  pure  benzene  and  p is  its 
partial  vapor  pressure  from  an  alcohol  solution  in  which  its 
molecular  fraction  is  x. 

2.  Ideal  Solutions. — We  have  discussed  the  character  of  the 
solutions  formed  when  benzene  is  shaken  successively  with  a 
series  of  liquids  of  continuously  increasing  resemblance  to  itself. 
There  remains  only  to  consider  the  limiting  case  of  the  solutions 
formed  by  mixing  with  benzene  a liquid  which  resembles  it  as 


Sec.  2] 


SOLUTIONS  III 


139 


closely  as  possible  and  whose  molecules  are,  therefore,  as  nearly 
like  those  of  benzene  as  possible.  Toluene  will  fulfill  these 
requirements  very  well  as  is  evident  from  a comparison  of  the 
formulas  of  the  two  molecules. 


Benzene 

Toluene , 

CH 

CH 

HC  CH 

HC  C.CH 

HC  CH 

HC  1^/  CH 

CH 

CH 

When  we  shake  these  two  non-associated  (Table  VI,  2)  liquids 
together,  we  find  that  they  not  only  mix  readily  with  each  other 
in  all  proportions,  but  we  note  also  that  the  process  of  mixing 
is  not  accompanied  by  any  appreciable  heat  effects  or  volume 
changes.  The  total  energy  of  the  system,  in  other  words,  is 
not  changed  when  the  two  liquids  are  mixed  together.  More 
over,  this  absence  of  energy  changes  is  found  to  be  generally  the 
case  whenever  any  two  very  similar  non-associated  liquids  are 
mixed  together  and  the  more  nearly  the  two  liquids  resemble 
each  other  the  more  exactly,  as  a rule,  does  this  relation  hold 
true.  This  behavior  is  interpreted  in  terms  of  our  concept  of 
thermodynamic  environment  by  the  view  that  the  thermody- 
namic environments  in  the  two  pure  liquids  are  practically  iden- 
tical, and  hence  on  mixing  the  liquids  there  is  no  change  in  this 
environment,  whatever  be  the  proportions  in  which  they  are 
mixed.2  That  is,  a benzene  molecule  (or  a toluene  molecule) 
finds  itself  in  practically  the  same  thermodynamic  environment, 
whether  it  be  surrounded  entirely  by  benzene  molecules,  or  en- 
tirely by  toluene  molecules,  or  by  mixtures  of  the  two  molecules 
in  any  proportions  whatever. 

Any  solution  in  which  the  thermodynamic  environment  is  con- 

2 The  process  of  mixing  in  such  a case  is  very  analogous  to  that  which 
occurs  when  two  perfect  gases  are  mixed  together  in  the  same  volume 
(XI,  3a).  The  free  energy  change  which  measures  the  magnitude  of  the 
change  in  thermodynamic  environment  can  be  calculated  from  equation  (1) 
above  and  has  always  been  found  equal  to  zero  when  the  two  liquids  resemble 
each  other  very  closely. 

10 


140 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIII 


stant,  and  entirely  independent  of  the  relative  proportions  of 
the  constituents  in  the  solution  will  be  called  an  ideal  solution. 
An  ideal  solution,  like  a perfect  gas,  is  strictly,  therefore,  only  a 
limiting  case  which  is  approached  the  more  closely,  the  more 
nearly  its  different  molecular  components  resemble  one  another. 
The  closest  approach  of  any  actual  solution  to  the  limiting  case 
of  an  ideal  solution  probably  occurs  when  we  mix  together  two 
liquid  hydrocarbons  which  are  optical  isomers  (I,  2d).  Such  a 
mixture  will  be  found  to  obey  all  of  the  laws  of  ideal  solutions  so 
closely  that  we  should  probably  be  unable  to  detect  the  slightest 
deviation  and  such  a solution  may,  therefore,  be  considered 
as  an  actual  example  of  an  ideal  solution.  It  will  be  readily 
seen  that  the  following  pairs  of  liquids  when  mixed  together 
will  also  form  solutions  which  will  be  very  close  to  ideal  solutions : 
(1)  chlorbenzene  and  brombenzene;  (2)  mercury  and  tin;  (3) 
krypton  and  xenon;  (4)  methyl  alcohol  and  ethyl  alcohol;  (5) 
holmium  and  dysprosium;  (6)  cesium  and  rubidium. 

3.  Dilute  Solutions. — In  general  a solution  composed  of  the 
two  substances,  A and  B,  will  have  a thermodynamic  environ- 
ment different  from  that  which  prevails  in  either  pure  liquid  A or 
pure  liquid  B and  this  thermodynamic  environment  will  vary 
with  changes  in  the  relative  amounts  of  A and  B in  the  solution. 
Suppose  we  take  any  solution  composed  of  A and  B and  pour 
into  it  some  pure  liquid  A.  This  process  is  called  “ diluting  the 
solution  with  A.”  As  the  mole-fraction  of  A increases  and  that 
of  B decreases,  the  thermodynamic  environment  in  the  solution 
approaches  gradually  that  which  prevails  in  pure  liquid  A and 
after  the  degree  of  dilution  has  become  great  enough  the  molecules 
of  B are  so  few  in  number  and  so  far  apart  that  their  influence 
upon  the  thermodynamic  environment  of  the  solution  becomes 
negligibly  small.  When  this  condition  is  reached,  further  dilution 
no  longer  produces  any  appreciable  change  in  the  thermodynamic 
environment.  In  other  words,  for  every  solution  in  which  one 
constituent,  the  solvent  (XI,  4)  largely  predominates  over  the 
other,  the  solute,  there  exists  a degree  of  dilution  beyond  which 
further  additions  of  solvent  no  longer  produce  any  appreciable 
effect  upon  the  thermodynamic  environment.  When  the  mole 
fraction  of  the  solvent  in  any  solution  is  so  large  that  the  thermo- 
dynamic environment  is  practically  identical  with  that  which 


Sec.  3] 


SOLUTIONS  III 


141 


prevails  in  the  pure  solvent,  the  solution  is  called  a “ dilute 
solution,”  or  more  accurately,  a “ sufficiently  dilute  solution.” 

The  exact  degree  of  dilution  which  the  solution  must  have 
before  its  thermodynamic  environment  becomes  practically  con- 
stant and  hence  independent  of  further  increases  in  the  mole  frac- 
tion of  thie  solvent  depends  upon  the  natures  of  the  solvent  and 
the  solute.  Strictly  speaking,  the  thermodynamic  environment 
never  reaches  absolute  constancy  until  the  solution  becomes 
infinitely  dilute,  but  for  practical  purposes  it  becomes  sufficient^ 
constant  at  moderate  dilutions  and  from  what  has  been  said 
above  with  reference  to  ideal  solutions,  it  is  evident  that  the  more 
closely  the  solvent  and  solute  resemble  each  other,  the  less  dilute 
does  the  solution  need  to  be  before  its  thermodynamic  environ- 
ment becomes  practically  constant.  No  very  accurate  general 
rules  can  be  formulated  for  deciding  just  how  dilute  a solution 
must  be  before  it  may  be  classed  as  a “ dilute  solution,”  because 
that  is  a question  which  can  only  be  decided  by  a study  of  the 
solution  itself.  From  the  knowledge  which  has  been  obtained 
from  the  study  of  the  behavior  of  aqueous  solutions,  how- 
ever, it  is  possible  to  formulate  the  following  general  rules 
which  hold  for  the  majority  of  cases  and  will  give  the  student 
a general  idea  of  the  concentration  range  which  is  usually  cov- 
ered by  the  term,  “ dilute  solution.” 

(a)  Aqueous  solutions  of  most  non-electrolytes  possess  a ther- 
modynamic environment  which  is  practically  constant  and  iden- 
tical (within  say  1 or  2 per  cent.)  with  that  which  prevails 
in  pure  water,  as  long  as  the  mole  fraction  of  the  solute  does 
not  exceed  about  0.01  (0.5  molal). 

(b)  In  a few  instances  (solutions  of  the  alcohols  and  the 
sugars)  this  limit  probably  extends  as  high  as  0.04  or  0.05,  while 
in  a few  others  it  is  very  possibly  considerably  lower  than  0.01. 

(c)  In  the  case  of  aqueous  solutions  of  strong  electrolytes, 
(1,  2g)  in  particular,  this  limit  must  be  placed  very  much  lower, 
as  low  as  0.00001  (0.0005  normal)  in  fact,  owing  to  the  powerful 
effect  which  ions  (1,  2g)  exert  upon  the  thermodynamic  environ- 
ment of  any  solution  containing  them. 

The  two  groups  of  solutions  which  we  have  just  considered, 
Ideal  Solutions  and  Sufficiently  Dilute  Solutions,  have  the 
common  characteristic  of  possessing  a thermodynamic  en- 


142 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIII 


vironment  which  does  not  change  with  variations  in  the  rela- 
tive amounts  of  the  constituents  of  the  solution,  and  both 
groups  of  solutions  obey  the  same  set  of  laws,  which  we  will 
call  the  Laws  of  Solutions  of  Constant  Thermodynamic  En- 
vironment and  which  we  shall  now  proceed  to  develop. 


REFERENCES 

Journal  Articles:  See  Trans.  Amer.  Electrocliem.  Soc.,  22,  333  (1912) 


CHAPTER  XIV 


SOLUTIONS  IV:  THE  LAWS  OF  SOLUTIONS  OF  CON- 
STANT THERMODYNAMIC  ENVIRONMENT. 


The  Distribution  Laws 


1.  The  General  Vapor  Pressure  Law. — For  solutions  in  which 
the  thermodynamic  environment  is  a constant  the  function 
fA  ( T.E .)  in  equation  (20,  XII)  is  constant  and  the  equation  may 
therefore  be  written 


(1) 


where  pA  is  the  partial  vapor  pressure  of  any  molecular  species, 
A,  from  a solution  in  which  its  molecular  fraction  (XI,  5)  is  xA, 
and  kA  is  a constant  characteristic  of  the  species,  A,  and  of  the 
thermodynamic  environment  which  surrounds  it  in  the  solution. 

Concerning  the  quantity,  pA,  in  this  equation  it  should  be 
remembered  that  the  directly  measured  vapor  pressure  above  a 
solution  is  determined  not  only  by  the  conditions  which  exist 
within  the  solution  but,  like  any  other  gas  pressure,  it  is  subject 
to  the  influences  which  exist  in  the  vapor  itself  and  which  at  high 
pressures  or  at  low  temperatures  cause  the  vapor  to  deviate 
appreciably  from  the  behavior  of  a perfect  gas  (II,  9),  If, 
therefore,  the  vapor  pressure,  pA,  in  equation  (1)  is  to  serve  as  a 
suitable  measure  of  the  escaping  tendency  of  the  molecular  spe- 
cies, A,  from  the  solution,  it  should,  strictly  speaking,  first  be 
corrected  for  those  influences  which  cause  the  vapor  to  deviate 
from  the  behavior  of  a perfect  gas.  This  “ corrected  vapor 
pressure”  has  been  called1  by  Lewis®  the  “fugacity”  of  the  mo- 

1 Lewis  [Proc.  Amer.  Acad.  Sci.,  37,  49  (1901)].  The  method  of  making 
the  correction  for  deviation  from  the  perfect  gas  law  is  discussed  in  this 
paper  and  also  in  a recent  paper  by  Gay  [Jour.  Chimie  Physique,  10,  197 
(1912)],  who  calls  the  corrected  vapor  pressure  the  “expansibility  tension” 
of  the  molecular  species  in  question. 

In  the  case  of  a distribution  law  involving  the  vapor  phase,  if  the  partial 

a Gilbert  Newton  Lewis  (1875).  Professor  of  Physical  Chemistry  and 
Dean  of  the  College  of  Chemistry  at  the  University  of  California. 


143 


144 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


lecular  species,  A,  in  the  solution.  It  is  evidently  a measure  of  the 
tendency  of  this  species  to  escape  from  the  thermodynamic  en- 
vironment, which  surrounds  it  in  the  solution,  into  that  condition 
of  zero  thermodynamic  environment  which  prevails  in  a perfect 
gas.  The  necessary  correction  which  must  be  applied  to  the 
observed  vapor  pressure  in  a given  case  in  order  to  obtain  the 
“fugacity,”  could  be  readily  calculated,  if  the  constants  of  the 
van  der  Waals’  or  of  the  Berthelot  equation  of  state  (II,  10a) 
were  known  for  the  vapor  in  question.  As  a matter  of  fact,  how- 
ever, in  many  cases  the  magnitude  of  this  correction  is  so  small 
that  it  falls  within  the  error  of  measurement  of  the  vapor  pressure 
itself  and  can,  therefore,  be  neglected.  In  general,  however, 
it  should  be  borne  in  mind  that  any  law  of  solution  which  in- 
volves the  vapor  phase  may  be  subject  to  deviations  of  the  same 
kind  and  order  of  magnitude  and  from  the  same  sources  as  those 
which  cause  gases  to  deviate  from  the  perfect  gas  law  (II,  10a). 

2.  The  Vapor  Pressure  of  Ideal  Solutions. — By  integrating 
equation  (1)  we  obtain 

Pa  = kAxA+I  (2) 

When  #A  = 0,  pA  = 0 also,  and  hence  the  integration  constant, 
I , is  zero.  When  = 1,  kA  = p0A,  the  vapor  pressure  of  pure  liquid 
A at  the  same  temperature  and  pressure.  Equation  (2),  there- 
fore, becomes: 

Pa  = Poaxa  (3) 

or  stated  in  words:  The  partial  vapor  pressure  of  any  molecu- 
lar species  above  an  ideal  solution  is  equal  to  its  vappr  pressure 
as  a pure  liquid  at  the  same  temperature  and  pressure  multi- 
plied by  its  molecular  fraction  in  the  solution.  In  other  words 
the  partial  vapor  pressure  of  any  molecular  species  from  an  ideal 
solution  is  a linear  function  of  its  molecular  fraction  in  that 
solution. 

vapor  pressure  employed  has  been  obtained  (as  is  usually  the  case)  by 
calculation  from  the  percentage  composition  of  the  vapor  in  equilibrium 
with  the  solution  on  the  assumption  that  the  vapor  behaves  like  a perfect 
gas,  the  vapor  pressure  so  calculated  is,  thereby,  usually  automatically 
“ corrected”  for  the  effect  of  the  influences  present  in  the  gas  phase  and  when 
substituted  in  the  distribution  law  in  question  usually  shows  good  agreement 
with  the  law.  This  is  the  case  with  the  data  exhibited  in  Fig.  19. 


Sec.  2 


SOLUTIONS  IV 


145 


Fig.  19. — The  Vapor  Pressure  Diagram  for  an  Ideal  Solution.  The  two 
intersecting  straight  lines  in  the  figure  are  the  graphs  of  the  theoretical 
equation,  using  the  values  of  po  indicated  on  the  right  and  left 

hand  margins  respectively.  The  upper  curve  is  the  theoretical  total  vapor 
pressure  curve.  The  small  circles  represent  the  observed  vapor  pressures 
at  85°  for  the  system  proplyene  bromide — ethylene  bromide  as  measured 
by  Zawidski.  [Z.  physik.  Chem.  35,  129  (1900)].  This  system  is  evidently 
very  close  to  an  ideal  solution  in  its  behavior. 


146 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


Problem  1. — Prove  that  the  total  vapor  pressure  above  an  ideal  solution 
made  up  of  the  two  molecular  species,  A and  B,  is  also  a linear  function  of 
the  molecular  fraction  of  each  species. 

The  vapor  pressure  law  and  its  application  to  an  actual  solu- 
tion are  illustrated  graphically  in  Fig.  19,  which  should  be  studied 
carefully  in  connection  with  equation  (3)  and  problem  1. 

3.  The  Vapor  Pressure  of  the  Solvent  from  a Dilute  Solution. 
Raoult’s  Law. — Integrating  equation  (1)  and  evaluating  the  con- 
stants in  the  same  manner  as  in  the  case  of  ideal  solutions,  we 
obtain  the  same  equation, 

p = Pox  (4) 


for  the  vapor  pressure  of  the  solvent  from  a dilute  solution. 

By  combination  with  equations  (2,  XI),  and  (9,  XII)  we  can  put 
this  in  the  form, 


Ap  Po~P  _ Vi 

Po  Po  N+N  i 


(5) 


where 


is  the  relative  vapor  pressure  lowering  produced  when 


P o 

N i moles  of  solute  are  dissolved  in  N moles  of  solvent. 

This  relation  is  known  as  Raoult’s  a law  of  vapor  pressure  lower- 
ing. In  words  it  states  that  the  relative  lowering  of  the  vapor 
pressure  of  a solvent,  which  occurs  when  a solute  is  dissolved  in 
it  to  form  a dilute  solution,  is  equal  to  the  mole  fraction  of  the 
solute  in  the  resulting  solution.  If  the  solute  is  a non-volatile 
one,  the  partial  vapor  pressure  of  the  solvent  is,  of  course,  also  the 
total  vapor  pressure  of  the  solution. 


Problem  2. — Solve  equation  (5)  so  as  to  obtain  an  expression  for  the 
molecular  weight,  Mi,  of  the  solute. 

Problem  3. — Show  that  for  very  dilute  solutions  Raoult’s  law  may  be 
put  in  the  form, 

(6) 

where  N_ i is  the  number  of  moles  of  solute  in  1000  grams  of  solvent  and  kp 
is  a constant  which  depends  only  upon  the  nature  of  the  solvent. 


For  solutions  which  are  so  dilute  that  equation  (6)  holds  with 
sufficient  accuracy,  the  relative  vapor  pressure  lowering  is  in- 
dependent of  the  temperature. 

a Frangois  Marie  Raoult  (1832-1901).  Professor  of  Chemistry  in  the 
Unversity  of  Grenoble.  Made  important  contributions  to  the  methods  of 
molecular  weight  determination  in  solution. 


Sec.  4] 


SOLUTIONS  IV 


147 


Problem  4. — Calculate  the  relative  vapor  pressure  lowering  produced 
when  one  formula  weight  of  diphenyl  (C4H10)  is  dissolved  in  1000  grams  of 
benzene. 

4.  The  Vapor  Pressure  of  the  Solute  from  a Dilute  Solution. 
Henry’s  Law. — To  obtain  an  expression  for  the  partial  vapor 
pressure,  pA,  of  any  solute,  A,  from  a dilute  solution,  we  have  only 
to  integrate  equation  (1)  again  which  gives 

pA  = kAxA+I  (7) 

If  xA  = 0,  pA  = 0 and  hence  1 = 0.  The  desired  relation  is,  therefore, 

pA  = kAxA  (8) 

where  xA  is  the  mole  fraction  of  the  solute  species,  A,  and  kA 
is  the  constant  characteristic  of  the  solute  A and  of  the  thermodynamic 
environment  which  surrounds  it  in  the  solution.  Equation  (8) 
is  known  as  Henry’s®  Law.  In  words  it  states  that  the  partial 
vapor  pressure  of  a solute  from  a dilute  solution  is  proportional 
to  its  mole  fraction  in  the  solution. 

Problem  6. — Show  that  if  the  solution  is  dilute  enough,  Henry’s  law  may 
be  put  in  either  of  the  following  forms: 

Px  = const.  xN_ a (9) 

pA  = const. XC  x (10) 

and 

C'A  = const. XC x and  x'A  = const. XxA  (11) 

where  A"a  and  Cx  are  the  weight-molal  and  volume-molal  concentrations 
(XI,  5),  respectively,  of  A in  the  solution  and  C'x  and  x'x  its  volume  con- 
centration and  mole  fraction  in  the  gas  above  the  solution. 

Table  XIV  illustrates  the  behavior  of  aqueous  solutions  of 

C' 

carbon  dioxide  toward  Henry’s  Law.  The  ratio,  is  usually 

called  the  11  solubility  of  the  gas  in  the  liquid Another  quantity 
known  as  the  li  absorption  coefficient ” of  the  gas  and  defined  by 
the  equation 

273.1  C'  (12) 

a-  TC 

where  T is  the  absolute  temperature,  is  frequently  employed  in 
recording  data  on  solubility  of  gases. 

a William  Henry,  F.  R.  S.  (1774-1836).  Son  of  the  chemist  Thomas 
Henry.  A physician  and  manufacturing  chemist  in  Manchester,  England. 
His  work  on  the  solubility  of  gases  was  published  in  1803. 


148 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


Table  XIV 

Partial  Pressure  and  Solubility  of  C02  in  Water  at  25°.  Illustrating 

Henry’s  Law,  /^yC°2  = const. 

CC02 

Measurements  by  Findlay,  Creighton,  and  Williams  [J.  Chem.  Soc.  97 
538  and  103,  637]. 


PC02, 

mm. 

265 

385 

485 

660 

760 

820 

955 

1060 

1150 

1240  : 

1350 

G'coi 

GCOi 

0.817 

0 

bo 

h-1 

CO 

'0.816 

0 

bo 

^4 

0 

00 

-4 

I 

0.816 

00 

d 

0.818 

0.818 

0.819 

0.820 

Problem  6. — Air  contains  0.04  per  cent,  by  volume  of  C02.  What  is  the 
concentration  of  carbon  dioxide  in  water  which  is  in  equilibrium  with  air 
at  25°  and  1 atm.?  (Use  Table  XIV.) 


5.  The  Distribution  Law  for  Dilute  Solutions.— If  we  take  a 
dilute  solution  of  some  substance  in  solvent,  1,  and  shake  this 
solution  with  another  solvent,  2,  with  which  the  first  solvent  is 
immiscible,  then  some  of  the  solute  molecules  will  pass  from 
the  first  to  the  second  solvent  until  finally  a state  of  equilibrium 
is  reached.  The  solute  is  now  said  to  be  in  distribution  equili- 
brium between  the  two  solvents  and  hence  according  to  the  Second 
Law  of  Thermodynamics  (see  the  last  paragraph  of  X,  10),  it 
must  have  the  same  vapor  pressure  from  both  solutions. 

Problem  7. — Prove  by  the  methods  of  Section  10,  Chapter  X,  that  the 
two  vapor  pressures  must  be  equal  in  such  a case. 


But  by  Henry’s  Law  (equation  8)  the  vapor  pressure,  p i,  of 
the  solute  molecules  above  solvent  1,  is 


V i = kiXi 

and  that,  p%,  above  solvent  2,  is 

7>2 = k 2X2 

and  since  Pi  = P2,  we  have 

X\  k 2 
x2~k  1 


- -I- 

— I —Kd 


(13) 

(14) 

(15) 


where  kD  is  the  so-called  distribution  coefficient  or  distribution 
constant  of  the  molecular  species  in  question  between  the  two 
solvents.  Stated  in  words:  When  any  molecular  species  in  dilute 
solution  is  in  distribution  equilibrium  between  two  immiscible 


Sec.  5] 


SOLUTIONS  IV 


149 

solvents  the  ratio  of  its  mole  fractions  in  the  two  solvents  is  al- 
ways equal  to  a constant  whose  value  is  characteristic  of  the  spe- 
cies in  question  and  of  the  two  thermodynamic  environments 
which  respectively  exist  in  the  two  solvents. 

Problem  8. — Show  that  if  the  two  mole  fractions  are  small  enough  equa- 
tion (15)  can  be  put  in  either  of  the  following  forms: 


Ni 

(16) 

,r  = const. 

n2 

c 

n = const. 

C2 

(17) 

where  H and  C are  respectively  the  wcight-molal  and  the  volume-molal 
concentrations  in  the  two  solvents. 

The  data  in  Table  XV  illustrate  the  behavior  of  HgBr2  in 
distribution  equilibrium  between  water  (W)  and  benzene  (B). 

Table  XV 

Distribution  of  IigBr2  between  water  and  benzene  at  25°.  Illustrating  the 
Distribution  Law,  C1/C2  =const.,  in  the  case  of  two  liquid  phases.  (Cf. 
XIV,  15.) 


Measurements  by  Sherrill  [Z.  phys.  Chem.  44,  70  (1903)]. 


C w = 

0.00320 

0.00634 

0.0115 

0.0170 

C B 

0.00353 

0.00715 

0.0130 

0.0194 

Cw/Cb  = 

0.906 

0.886 

0.880 

0.876 

Although  in  the  above  statement  and  derivation  of  the  dis- 
tribution law  we  have  employed  liquid  solutions,  the  law  is  a more 
general  one,  one  in  fact  which  governs  the  distribution  equilibrium 
of  any  molecular  species  between  any  two  phases  whatever,  be 
they  crystalline,  liquid  or  gaseous,  provided  only  that  the 
thermodynamic  environment  in  each  phase  does  not  change  with 
the  mole  fraction  of  the  solute  in  that  phase,  that  is,  provided  that 
the  solution  in  each  phase  is  a ‘‘sufficiently  dilute  solution” 
(XIII,  3) . Henry’s  law  as  expressed  by  equation  ( 1 1 ) is  evidently 
only  a special  case  of  this  general  distribution  law. 

Problem  9.  — At  25°  the  distribution  ratio  of  Br2  between  carbon  tetra- 
chloride and  water  is  ^^  = 38.  The  partial  pressure  of  Br2  above  a 0.05 

molal  solution  of  it  in  water  at  25°  is  50  mm.  If  1 liter  of  this  solution  is 
shaken  with  50  c.c.  of  carbon  tetrachloride,  what  will  be  the  pressure  of  the 
bromine  above  the  carbon  tetrachloride  phase. 


150 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


Problem  10. — At  25°  the  solubility  of  iodine,  I2,  in  CC14  is  30.33  grams  per 
liter  and  in  water  0.001341  moles  per  liter.  One  liter  of  CC14  containing 
25  grams  of  I2  is  shaken  with  3 liters  of  water  at  25°  until  equilibrium  is 
established.  How  many  grams  of  iodine  will  be  found  in  the  water  layer? 


Osmotic  Pressure  and  Osmosis3 


6.  The  General  Osmotic  Pressure  Law. — By  dividing  equation 
(1)  by  its  integral,  pA  = A;A:rA,  we  eliminate  the  constant,  kA,  and 
obtain 

d pA  = pA  J (18) 

xA 

By  combining  this  equation  with  equation  (15,  XII)  so  as  to  elimi- 
nate dp,  and  putting  RT  in  place  of  pv o in  the  result  we  have 
the  osmotic  pressure  law  for  solutions  of  constant  thermody- 
namic environment,  namely, 


dnA  = 


— RT  dxA 
V oA  xA 


- RT 

PoA 


d log,  xA 


(19) 


This  expresses  the  osmotic  pressure  of  the  solution  referred  to 
constituent,  A,  (XII,  7)  whose  molal  volume  as  a pure  liquid  is 
VoA  and  whose  mole  fraction  in  the  solution  is  xA. 

If  we  assume  that  the  substance  A as  a pure  liquid  is  incom- 
pressible, then  P0A  is  a constant  independent  of  n and  the  gen- 
eral integral  of  equation  (19)  is 


nA=-~logexA  (20) 

V 0 A 

the  integration  constant  being  obviously  zero.  This  equation 
expresses  the  general  connection  between  the  osmotic  pressure  of 
an  ideal  solution  and  its  composition. 

The  same  equation  also  applies  to  dilute  solutions,  if  we 
understand  that  the  solvent  is  the  constituent  which  is  present 
as  the  pure  liquid  (cf.  the  definition  of  osmotic  pressure,  XII, 
7),  and  with  this  understanding  we  may  drop  the  subscript,  A, 
and  write 

RT  , 

n = - log,  x (21) 

v 0 


x being  the  mole  fraction  of  the  solvent  in  the  solution  and  V o 
its  molal  volume  as  a pure  liquid  at  the  temperature  T. 


Sec.  7] 


SOLUTIONS  IV 


151 


7.  Direct  Osmotic  Pressure  Measurements. — The  pressure 
difference  necessary  to  prevent  osmosis  through  a perfectly  semi- 
permeable  (XII,  7)  membrane  is  equal  to  the  osmotic  pressure 
and  by  determining  this  pressure  difference  for  a given  solution 
the  osmotic  pressure  is  thus  directly  measured.  Membranes 
which  approximate  perfect  semipermeability  have  thus  far  been 
obtained  only  in  the  case  of  a few  aqueous  sugat  solutions  and 
accurate  measurements  of  osmotic  pressure  by  this  method  are 
consequently  very  few  in  number. 

In  Table  XVI  below  are  shown  some  results  obtained  by  Morse® 
and  his  associates  for  the  osmotic  pressures  of  cane-sugar  solu- 
tions at  25°.  These  investigators  employed  membranes  of  cupric 
ferrocyanide  deposited  in  the  pores  of  a porous  earthenware  cell. 
The  sugar  solution  was  placed  in  the  interior  of  the  cell  which 
was  then  surrounded  by  water.  The  pressure  upon  the  solution 
was  then  increased  until  it  was  just  sufficient  to  prevent  the 
passage  of  water  into  or  out  of  the  cell  through  the  cupric 
ferrocyanide  membrane. 

It  is  not  without  interest  to  compare  the  values  of  the  osmotic 
pressure  obtained  in  this  way  with  those  calculated  from  the 
theoretical  equation  for  an  ideal  solution  (21),  although  for  the 
solutions  employed  by  Morse  the  calculation  cannot  be  carried 
out  without  making  several  of  assumptions.  In  the  first  place 
the  osmotic  pressure  measured  by  Morse  is  that  defined  by 
equation  (13,  XII)  while  that  which  appears  in  the  equation  for 
an  ideal  solution  (21)  is  defined  by  equation  (12,  XII).  For 
cane  sugar  solutions  at  25°  however,  the  two  quantities  II 
and  II'  do  not  differ  by  more  than  0.1  per  cent,  for  the  concen- 
tration range  covered  by  table.  Another  difficulty  in  applying 
equation  (21)  to  these  solutions  is  that  this  equation  assumes  that 
the  “pure  solvent,”  that  is,  the  constituent  to  which  the  mem- 
brane is  permeable,  is  composed  of  only  one  species  of  molecule, 
a condition  which  is  not  fufilled  in  the  case  of  water.  A third 
difficulty  lies  in  the  fact  that  for  cane-sugar  solutions  the  molecu- 
lar formula  of  the  solute  in  the  solution  is  not  known  with  cer- 
tainty, for  the  sugar  molecules  very  probably  become  hydrated 
when  they  dissolve  in  water. 

a Harmon  Northrup  Morse  (1848-  ).  Professor  of  Inorganic  and 

Analytical  Chemistry  at  Johns  Hopkins  University. 


152 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


Table  XVI 

Comparison  of  the  measured  values  of  the  osmotic  pressure,  n',  of  aqueous 
cane-sugar  solutions  at  25°  with  the  values  calculated  from  the  relation, 
_-RT  N _ -0.08207  X (273.1 +25)  X2.303  N 

n_  Vo  °ge  N+Ni~  1.00294X18.015  l0gl°  N+N\ 

1000 


Measurements  by  Morse  and  Associates,  Amer.  Chem.  Jour.  45,  600  (1911) 


Weight  formal 
concentration  of 
C12H22O11 

Ni 

Observed  osmotic 
pressure  (in 
atmospheres) 

n' 

Calculated  os- 
motic pressure 
assuming  the 
formula 
C12H22O11 
lb 

Calculated  os- 
motic pressure 
assuming  the 
formula 

C12H22O11.6H2O 

n2 

0.09924 

2.63 

2.4 

2.4 

0.1985 

5.15 

4.8 

4.9 

0.2978 

7.73 

7.2 

7.5 

0.4962 

12.94 

12.0 

12.7 

0.5954 

15.62 

14.4 

15.4 

0.6946 

18.43 

16.8 

18.2 

0.7929 

21.25 

19.2 

21.1 

0.8931 

24.13 

21.6 

23.9 

0.9924 

27.05 

24.0 

26.7 

If  it  be  assumed  that  the  sugar  is  not  hydrated  and  that 
water  has  the  molecular  weight  18,  then  the  mole  fraction  of  the 
water  in  a solution  containing  N i moles  of  sugar  (C12H22O11) 
per  1000  grams  of  water  would  be 


x — 


1000^ 

18 


(22) 


If  values  of  x calculated  in  this  way  be  substituted  in  the  theo- 
retical equation  (21),  the  values  of  II 1 thus  obtained  are  shown 
in  column  3 of  Table  XVI. 

These  values  do  not  agree  very  well  with  the  experimental 
ones.  Part  of  the  disagreement  is  undoubtedly  due  to  the  fact 
that  water  is  not  a pure  substance  with  a molecular  weight  in 
the  liquid  state  of  18.  The  disagreement  is  probably  too  great 
to  be  accounted  for  by  this  fact  alone,  however,  for  the  value 
assumed  for  the  molecular  weight  of  water  appears  in  the  equation 


Sec.  7] 


SOLUTIONS  IV 


153 


in  such  a way  that  for  moderate  concentrations  it  cancels  itself 
out  approximately,  as  can  be  seen  by  a careful  examination  of 
equations  (21)  and  (22).  Most  of  the  difference  between  the 
calculated  and  the  observed  values  can  probably  be  attributed  to 
an  erroneous  assumption  with  regard  to  the  molecular  weight  of 
sugar  when  dissolved  in  water.  Instead  of  remaining  unhy- 
drated, it  is  probable  that  the  sugar  molecules  unite  with  some 
of  the  water  to  form  a hydrate.  If  the  number  of  water  mole- 
cules which  unite  in  this  way  with  one  molecule  of  sugar  is  W, 
then  the  mole  fraction  of  the  free  water  as  computed  from  equa- 
tion (22)  above  would  evidently  be 

x = ("is"  “ W &)  + (is"  - (23) 

If  we  assume  W = 6 and  substitute  values  of  x computed  from 
equation  (23)  in  equation  (21),  we  obtain  the  values  of  n2 
given  in  the  column  4 of  Table  XVI.  These  values  show  good 
agreement  with  the  experimental  ones  and  this  agreement  may 
be  taken  as  evidence  that  the  molecules  of  cane  sugar  in  aqueous 
solution  are  on  the  average  each  hydrated  with  approximately  6 
molecules  of  water.  The  molecular  formula  of  the  solute  in 
solution  would,  therefore,  be  C12H22O116H2O. 

We  have  mentioned  the  fact  (XII,  7)  that  the  osmotic  pressure 
of  any  solution  can  be  thermodynamically  calculated  from  the 
other  colligative  properties  of  the  solution.  As  an  example  of 
the  values  thus  obtained  in  the  case  of  cane  sugar  solutions  at 
0°,  Table  XVII  is  given.  The  “ calculated  ” values  were  computed 
from  the  freezing  points  of  the  solutions  by  means  of  the  integral 
of  equation  (17,  XII).  The  “ observed ” values  were  directly 
measured  by  Morse  and  his  associates. 

This  problem  illustrates  the  application  of  the  osmotic  pressure 
law  to  the  determination  of  the  molecular  weight  of  substances 
whose  molecules  are  made  up  of  a very  large  number  of  atoms. 
Molecules  such  as  those  of  dextrin  and  albumin  are,  in  fact,  so 
large  that  solutions  of  these  substances  resemble  colloidal  solu- 
tions in  many  respects  and  are  usually  classed  with  such  solu- 
tions for  the  purposes  of  systematic  treatment.  Owing  to  the 
small  molal  concentration  of  many  of  these  solutions,  the  freezing 
point  lowering  is  so  small  that  it  cannot  be  measured  with  suffi- 


154 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


Table  XVII 

Comparison  of  the  osmotic  pressures  of  cane  sugar  solutions  at  0°  deter- 
mined by  direct  measurement,  with  the  osmotic  pressures  thermodynamic- 
ally calculated  from  the  freezing  point  lowerings  by  means  of  the  integral 
of  equation  (17,  XII)  which  for  aqueous  solutions  may  be  written, 
II  = 12.06  (Af/r  - 1.78  X 10-3A^2  -2.5  X 10~6A^3). 

Measurements  by  Morse  and  Associates,  Amer.  Chem.  Jour.  45,  600 
(1911).  Calculations  according  to  Lewis,  Jour.  Amer.  Chem.  Soc.  30,  671 
(1908). 


Weight-formal 
concentration  of 
C12H22O11 

V, 

Freezing 

point 

lowering 

AtF 

Observed 

osmotic 

pressure 

n' 

1 Calculated 
osmotic 
pressure 

n 

0709924 

0.195 

(2.46) 

2.35 

0. 1985 

0.392 

4.72 

4.73 

0.2978 

0.585 

7.08 

7.05 

0.4962 

0.985 

11.9 

11.9 

0.5954 

1.19 

14.4 

14.4 

0.6946 

1.39 

16.9 

16.8 

0.7929 

1.62 

19.5 

19.5 

0.8931 

1.83 

22.1 

22.0 

0.9924 

2.07 

24.8 

24.9 

Problem  9. — A 1 per  cent,  solution  of  dextrin  in  water  has  at  16°  an  os- 
motic pressure  of  16.6  cm.  of  mercury.  Calculate  the  molecular  weight  of 
the  solute.  (Use  the  van’t  Hoff  equation  (28).)  To  what  fraction  of  a 
degree  would  it  be  necessary  to  know  the  freezing  point  of  this  solution  in 
order  to  determine  the  molecular  weight  of  dextrin  to  1 per  cent,  by  the 
freezing  point  method?  (Use  equation  (51).) 

cient  accuracy  to  give  a satisfactory  value  for  the  molecular 
weight  of  the  solute.  Moreover  the  freezing  point  is  liable  to  be 
seriously  affected  by  the  presence  of  traces  of  impurities.  It  is 
with  such  solutions  as  these  that  the  osmotic  pressure  method 
is  of  particular  value  for  molecular  weight  determinations. 
Membranes  quite  impermeable  to  the  large  solute  molecules,  and 
sufficiently  strong  to  withstand  the  small  pressures  involved,  are 
comparatively  easy  to  secure  and  these  membranes  are  usually 
permeable  to  many  of  the  impurities  present  in  the  solution  and 
hence  the  measured  osmotic  pressure  is  not  influenced  by  these 
impurities. 


Sec.  8] 


SOLUTIONS  IV 


155 


8.  The  Van’t  Hoff  Equation. — In  the  comparisons  given  in 
Table  XVI  we  employed  the  general  osmotic  pressure  law  in  the 
form, 

U=-~~\ogex  (24) 

v 0 

If  the  mole  fraction  of  the  solute  is  small  enough,  however, 
this  equation  may  be  rearranged  into  a more  convenient  form  for 
calculation.  In  place  of  x we  may  put  1 —Xi  (2,  XI),  where  X\ 
is  the  mole  fraction  of  the  solute  in  the  solution.  This  gives, 

n=-^log.  (l-*i)  (25) 

v 0 

and  by  expanding  the  logarithm  into  a series  with  the  help  of 
Maclaurin’s  Theorem  we  have 


n 


RT 


+ + ) 


(26) 


If  the  mole  fraction  of  the  solute  is  small  enough,  its  higher  powers 
may  be  neglected  in  comparison  with  its  first  power  and  we 
can  write 


_RT  RT  N i 
1 V„Xl  Vo  JV+iVi 


(27) 


and  if  we  may  neglect  N i in  comparison  N in  the  denominator  of 
the  right-hand  member  and  write  V in  place  of  NV0,  this  equation 
becomes 

UV  = NiRT  (28) 


where  Ni  is  evidently  the  number  of  moles  of  solute  in  V liters 
of  solvent  at  T°.  As  the  solution  becomes  more  and  more 
dilute  V becomes  practically  identical  with  the  volume,  Vs,  of 
the  solution  containing  N i moles  of  solute.  When  this  is 
true  we  have 

UVS  = N1RT  or  n = Ci^T  (29) 

The  formal  resemblance  between  equation  (29)  and  the  perfect 
gas  law  is  obvious  and  for  this  reason  it  is  possible  to  say  that 
the  osmotic  pressure  of  an  extremely  dilute  solution  is  numeric- 
ally equal  to  the  pressure  which  the  solute  molecules  would 
exert  as  a gas,  if  they  occupied  the  same  volume  as  they  do  in  the 
solution  and  had  the  same  temperature.  This  is  the  form  in 
which  the  osmotic  prssure  law  was  first  stated  by  van’t  Hoff  in 

li 


156  PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 

1886  and  equation  (29)  is  known  as  van’t  Hoff’s  law  of  osmotic 
pressure.  Now  the  pressure  of  a gas  is  due  to  the  impacts  of  the 
molecules  of  the  gas  upon  the  walls  of  the  containing  vessel  and 
owing  to  the  formal  resemblance  between  van’t  Hoff’s  osmotic 
pressure  law  and  the  equation  of  state  of  a perfect  gas,  many 

persons  have  held  the  view  that  the 
osmotic  pressure  of  a solution  is  a real 
pressure  existing  within  the  solution  and 
caused  by  the  impacts  of  the  solute 
molecules  against  any  retaining  surface. 
In  order  to  make  clear  the  illogical  char- 
acter of  such  a view  we  will  consider  in 
some  detail  the  molecular  kinetic  inter- 
pretation of  the  process  of  osmosis. 

9.  The  Molecular  Kinetic  Interpreta- 
tion of  the  Process  of  Osmosis.  (a) 
Osmosis  in  the  Case  of  Ideal  Solutions. 
— Fig.  20  represents  two  vessels  each 
filled  with  a liquid  A which  for  con- 
venience we  will  designate  as  our  solvent. 
Each  vessel  is  provided  with  a piston  by 
means  of  which  the  total  pressure  upon 
the  liquid  may  be  varied  at  will.  The 
two  vessels  are  connected  by  a tube  fitted 
with  a membrane,  M,  of  infinitesimal 
thickness  which  allows  the  molecules  of 
A to  pass  through  it,  but  which  is  impermeable  to  every  other 
species  of  molecule.  Let  both  vessels  be  initially  under  the 
same  pressure  and  at  the  same  temperature. 

The  molecules  of  A being  in  constant  motion  will  strike  the 
membrane,  M,  from  both  sides  and  hence  will  pass  through  it  in 
both  directions,  but  since  the  concentrations  of  the  A molecules 
and  the  temperature,  pressure  and  other  conditions  are  the  same 
in  both  vessels,  equal  numbers  of  molecules  will  strike  both  sides 
of  the  membrane  in  unit  time  and  hence  the  quantities  of  A pass- 
ing through  the  membrane  will  be  the  same  in  the  two  directions. 
Suppose  now  that  we  dissolve  in  the  liquid  in  the  right-hand 
vessel  some  substance,  B,  which  forms  an  ideal  solution  with  A. 
Since  there  are  now  fewer  molecules  of  A per  unit  volume  in  the 


Fig.  20. 


Sec.  9] 


SOLUTIONS  IV 


157 


right-hand  vessel,  the  number  of  these  molecules  which  can 
strike  the  semipermeable  membrane  from  the  right  in  unit  time 
must  be  less  than  the  number  which  strike  it  from  the  left  and 
hence  there  must  be  a flow  of  A from  left  to  right  through  the 
membrane.  In  other  words  osmosis  must  occur. 

This  osmosis  might  be  stopped  by  (1)  raising  the  temperature 
of  the  right-hand  vessel  or  by  lowering  that  of  the  left-hand 
vessel  or  it  might  also  be  stopped  by  (2)  raising  the  pressure  on 
the  right-hand  vessel  or  by  lowering  that  on  the  left-hand  vessel. 
If  the  second  method  is  employed  the  pressure  difference  neces- 
sary to  make  the  rates  of  passage  of  the  A molecules  through  the 
membrane  the  same  in  the  two  directions,  is  evidently  what  we 
have  defined  (XII,  7)  as  the  osmotic  pressure  of  the  solution.2 
One  effect  of  increasing  the  pressure  upon  any  liquid  is  to  force 
the  molecules  closer  together  and,  therefore,  to  increase  the 
repulsive  forces  acting  between  them  and  hence  to  increase  the 
vigor  of  their  molecular  impacts  against  any  surface.  By 
increasing  the  pressure  upon  the  right-hand  vessel,  therefore, 
the  vigor  of  the  impacts  of  the  A molecules  against  the  semi- 
permeable membrane  (and  to  a slight  extent  the  concentration 
of  these  molecules  as  well)  would  be  increased  and  the  number 
of  molecules  which  are  able  to  pass  through  the  membrane  per 
unit  time  would  consequently  be  increased.  This  is  the  most 
probable  molecular  kinetic  interpretation  of  the  process  of 
osmosis  and  of  the  way  in  which  the  change  of  pressure  on  one 
side  of  the  membrane  brings  the  osmosis  to  a stop,  in  the  simple 
case  of  an  ideal  solution. 

As  the  mole  fraction  of  the  B molecules  in  the  right-hand 
vessel  of  Fig.  20  is  increased  the  osmotic  pressure  increases  in 
accordance  with  the  integral  of  equation  (19)  and,  as  this  in- 
tegral shows,  the  osmotic  pressure  must  approach  infinity  as 
the  mole  fraction  of  B approaches  unity.  This  statement 
holds  true  regardless  of  the  natures  of  A and  B.  In  other 
words  no  pressure  difference,  however  great,  would  be  able 
to  prevent  the  passage  of  A molecules  from  pure  A on  the 
left  of  the  membrane  into  pure  B on  the  right. 

2 Evidently  an  “osmotic  temperature”  for  the  solution  might  be  similarly 
defined,  if  one  wished  to  do  so. 


158 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


(b)  Osmosis  in  General. — In  the  simple  case  of  the  ideal  solu- 
tion just  considered  we  have  seen  that  the  solvent  would  flow 
through  the  semipermeable  membrane  into  the  solution  merely 
because  there  is  a greater  number  of  solvent  molecules  per 
unit  volume  in  the  pure  solvent  than  in  the  solution  and  hence 
the  number  of  molecules  striking  the  membrane  from  the  sol- 
vent side  will  naturally  be  the  greater.  In  other  words  the  os- 
mosis in  such  a case  would  be  a purely  kinetic  phenomenon  and 
quite  analogous  to  the  passage  of  hydrogen  gas  through  a 
palladium  wall  into  a gaseous  mixture  containing  hydrogen  at  a 
smaller  concentration  (Cf.  II,  6).  In  general,  however,  the 
phenomenon  of  osmosis  is  a more  complex  process,  one  which  is 
probably  quite  appreciably  influenced  by  the  attractive  forces 
(neutralized  in  the  cases  of  ideal  solutions)  which  act  between 
the  molecules  in  the  interior  of  the  liquid.  Thus  when  a strong 
aqueous  sugar  solution  is  separated  from  pure  water  by  a mem- 
brane permeable  only  to  the  water,  the  rate  of  osmosis  might 
be  very  appreciably  influenced  by  attractive  forces  acting  be- 
tween the  sugar  molecules  on  one  side  of  the  membrane  and 
the  water  molecules  on  the  other  side  of  the  membrane,  or  in  the 
interior  of  the  membrane  if  the  latter  is  of  appreciable  thickness. 
The  presence  of  such  forces  would  affect  also  the  pressure  differ- 
ence which  must  be  established  on  solvent  and  solution  respec- 
tively in  order  to  stop  the  osmosis.  This  is  only  another  way  of 
stating  that  the  osmotic  pressure  of  a solution  is  in  general  a 
function  of  the  thermodynamic  environment  prevailing  within 
the  solution.  (Cf.  XIII,  1.) 

In  the  above  interpretation  of  the  process  of  osmosis  it  will 
be  noticed  that  we  have  said  nothing  about  any  pressure  exerted 
against  the  semipermeable  membrane  by  the  molecules  of  the  solute, 
nor  is  it  necessary  that  we  should.  Such  a pressure  would  un- 
doubtedly exist,  for  the  membrane  is  constantly  being  bombarded 
by  these  solute  molecules,  but  this  pressure  is  not  the  osmotic 
pressure  of  the  solution.  The  osmotic  pressure  of  a solution  is, 
in  fact,  not  a real  pressure  exerted  by  something  within  solution, 
but  is  instead  an  abstraction,  representing  simply  the  pressure 
difference  which  would  have  to  be  established  upon  solution  and 
pure  solvent  respectively  in  order  that  the  solvent  should  have 
the  same  escaping  tendency  from  both  of  them.  It  is  thus  a 


Sec.  9] 


SOLUTIONS  IV 


159 


definite  physical  quantity  quite  independent  of  semipermeable 
membranes,  osmosis  or  molecular  theory  of  liquids.  The  partial 
pressure  of  the  solute  molecules  against  the  semipermeable 
membrane  has,  however,  been  frequently  given  as  the  cause  of 
osmosis  and  has  been  identified  with  the  osmotic  pressure.  To 
what  extent  this  is  justifiable  will  become  evident  from  the 
following  considerations. 

(c)  Thermal  Pressure  and  Diffusion  Pressure. — When  a dilute 
solution  of  B in  A in  the  right-hand  vessel  of  Fig.  20  is  separated 
from  pure  A in  the  left-hand  vessel  by  a membrane  permeable 
only  to  A,  the  molecules  of  B which  are  constantly  striking  the 
right-hand  side  of  this  membrane  will  exert  against  it  a certain 
partial  pressure,  pD,  which  we  will  call  the  diffusion  pressure  of 
these  molecules,  since  it  is  evidently  a measure  of  the  tendency 
of  the  solute  to  diffuse  into  the  pure  solvent,  the  diffusion  being 
in  the  above  case  prevented  by  the  presence  of  the  membrane. 
The  following  molecular  kinetic  analysis  of  the  conditions  at  the 
surface  of  the  membrane  will  aid  us  in  appreciating  the  nature  of 
some  of  the  factors  which  determine  the  magnitude  of  this  diffu- 
sion pressure  of  the  solute  molecules. 

Let  both  liquids  be  under  atmospheric  pressure  and  consider 
the  layer  of  B molecules  which  at  any  moment  are  just  about 
to  strike  the  membrane  from  the  right.  If  these  molecules  were 
not  subject  to  any  attractive  forces  from  their  neighbors,  they 
would  exert  against  the  membrane  a pressure,  pT,  which  will  be 
called  their  thermal  pressure  and  which  would  be  equal  to  the 
pressure  exerted  by  an  equal  number  of  molecules  of  a gas  at 
the  same  temperature  and  the  same  effective  concentration.  As 
a matter  of  fact,  however,  the  layer  of  B molecules  next  the 
membrane  will  be  subject  to  a pull  backward  by  both  the  A 
and  the  B molecules  which  are  behind  them  and  to  a pull  forward 
by  the  A molecules  on  the  other  side  of  the  membrane.  These 
different  attractive  forces  will  influence  the  force  with  which  the  B 
molecules,  strike  the  membrane,  and  will  thus  be  a determining 
factor  in  the  effective  pressure  which  these  molecules  will  be 
able  to  exert  against  the  membrane.  If  we  call  pA  and  pB  the 
backward  pulls  (in  pressure  units)  exerted  by  the  A and  the  B 
molecules  in  the  solution,  and  p'A  the  forward  pull  exerted  by  the 
A molecules  on  the  other  side  of  the  membrane,  then  the  actual 


160 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


pressure,  pD,  which  the  B molecules  will  exert  against  the  mem- 
brane, when  they  strike  it,  will  be 

Pd  = Pt+p'a~(Pa+Pb ) (30) 

For  the  thermal  pressure,  pT,  we  might  write 

Pr  =jyz^)  = cCot.  RT  (31) 

where  CCor.  is  the  molal  concentration  of  the  B molecules,  cor- 
rected for  the  space  actually  filled  by  these  molecules,  the 
correction  being  made  according  to  the  method  of  reasoning 
employed  by  van  der  Waals  in  correcting  the  gas  law  for  high 
pressures.  (Cf.  II,  10a.)  CCor.  will  be  called  the  effective  con- 
centration of  the  B molecules. 

In  the  limiting  case  of  an  ideal  solution  p'a  = Pa+Pb  for  all 
concentrations  and  in  the  case  of  any  dilute  solution,  as  the 
concentration  of  the  solute,  B,  decreases  and  approaches  zero, 
pB  approaches  zero  and  p'A  approaches  Pa,  so  that  here  also  p\ 
= Pa+Pb-  In  words,  the  diffusion  pressure  of  the  molecules  of 
the  solute  in  an  ideal  solution  or  in  an  infinitely  dilute  solution 
is  equal  to  their  thermal  pressure  and  hence  is  equal  to  the  pres- 
sure which  they  would  exert  in  the  gaseous  state  at  the  same 
temperature  and  the  same  effective  concentration. 

If  we  start  with  a solution  in  which  the  concentration  of  the 
solute  is  small,  then  as  we  gradually  increase  it,  the  value  of  p# 
(equation  30)  will  evidently  always  remain  finite  and,  if  the 
solution  is  an  ideal  one,  PD  = PT  = Cc0r.RT  for  all  values  of  the 
concentration.  If  the  solution  is  not  an  ideal  solution,  the  value 
of  pD  is  equal  to  CRT  for  small  concentrations,  but  as  the  con- 
centration of  the  solute  increases,  pD  will  at  first  increase  more 
slowly  than  CCorRT,  owing  to  the  fact  that  Pa+Pb  m equation 
(30)  is  greater  than  p'A.  As  the  concentration  of  the  solute  con- 
tinues to  increase,  however,  a point  may  be  reached  where  p\  is 
greater  than  Pa+Pb-  If  this  is  true,  the  value  of  pD  might  for  a 
time  increase  faster  than  CCorRT . In  other  words  the  variation 
of  pD  with  the  concentration  is  in  general  not  predictable  between 
infinite  dilution  and  pure  solute.  When  the  mole  fraction  of  the 
solute  in  the  right-hand  vessel  becomes  unity,  however,  pA  obvious- 
ly becomes  zero  and  we  have,  pD  =Pt’+p'a  — Pb-  Beyond  the 


Sec.  9] 


SOLUTIONS  IV 


161 


fact  that  pD  under  these  conditions  must  still  be  a finite  quantity, 
we  cannot  make  any  general  statement  with  regard  to  its  magni- 
tude or  even  with  regard  to  its  sign,  although  these  can  usually 
be  determined  in  any  specific  case. 

In  the  preceding  discussion  we  have  considered  the  conditions 
which  exist  on  the  two  sides  of  the  membrane  when  both  the  solu- 
tion and  the  pure  solvent  are  under  the  same  total  pressure  and 
at  the  same  temperature.  If  the  pressure  or  temperature  of 
the  solution  be  altered,  however,  there  will  be  corresponding 
changes  in  the  thermal  pressure  and  the  diffusion  pressure 
of  the  solute  molecules.  Increase  of  pressure  upon  the  solution 
will  cause  a corresponding  increase  in  the  diffusion  pressure  of 
the  solute  molecules  and  their  diffusion  pressure  can  be  made 
numerically  equal  to  the  osmotic  pressure,  II'  defined  by  equation 
(13),  by  putting  the  solution  under  the  pressure,  P,  also  defined 
by  that  equation.  In  other  words  for  the  special  case  when  a 
solution  is  in  osmotic  equilibrium  with  the  pure  solvent,  the 
diffusion  pressure  of  the  solute  molecules  is  equal  in  magnitude 
to  the  osmotic  pressure,  II',  of  the  solution. 

(d)  Summary.1 — We  may  sum  up  our  discussion  in  the  following 
terms.  Osmotic  pressure,  thermal  pressure,  and  diffusion  pres- 
sure are  three  quite  distinct  quantities  and  should  be  recognized 
as  such.  Briefly  they  are  defined  and  described  as  follows: 

(1)  Osmotic  Pressure. — The  osmotic  pressure  of  a solution  is 
the  pressure  difference  which  must  be  established  upon  the  solu- 
tion and  the  pure  solvent  respectively,  in  order  to  make  the 
escaping  tendency  of  the  solvent  the  same  from  both  of  them. 
The  osmotic  pressure  is,  therefore,  not  a real  pressure  existing 
within  the  solution  but  is  a definite  physical  quantity  quite  in- 
dependent of  osmosis,  semipermeable  membranes  or  molecular 
theory.  It  is  connected  with  the  other  colligative  properties 
of  the  solution  by  definite  relations  which  can  be  deduced  by 
purely  thermodynamic  reasoning.  For  dilute  solutions  the 
osmotic  pressure  becomes  in  the  limit  equal  to  CRT  where  C 
is  the  concentration  of  the  solute,  but  as  the  concentration  in- 
creases the  osmotic  pressure  approaches  infinity  as  its  upper  limit. 

(2)  Thermal  Pressure. — Every  molecular  species  in  a solution 
possesses  the  unordered  heat  motion  of  all  fluid  molecules  and  by 
virtue  of  this  motion  its  molecules  may  be  considered  as  having  a 


162  PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 

corresponding  pressure,  pT,  called  their  thermal  pressure.  For 
dilute  solutions  of  the  species  in  question  this  pressure  will  be 
equal  to  CRT  but  as  the  concentration  of  the  molecular  species 
increases  the  value  of  pT  also  increases  and  approaches  a finite 
hut  usually  very  large  upper  limit,  the  thermal  pressure  of  the 
molecules  in  the  pure  liquid  solute.  If  van  der  Waals’  equation 
held  for  the  liquid,  the  thermal  pressure  would  be  equal  to 

NiRT 

(V-h) 

(3)  Diffusion  Pressure. — The  partial  pressure  which  the  solute 
molecules  in  any  solution  would  exert  against  a membrane  per- 
meable only  to  the  solvent  is  called  their  diffusion  pressure,  pD. 
For  dilute  solutions  the  diffusion  pressure  is  equal  to  CRT 
but  as  the  concentration  of  the  solute  increases,  the  diffusion 
pressure  remains  finite  for  all  values  of  C and  reaches  a definite 
limiting  value  whose  magnitude  and  sign  depend  upon  the  tem- 
perature, the  external  pressure  and  the  attractive  forces  which 
are  acting  in  the  interior  of  the  liquid. 

The  way  in  which  the  three  quantities,  II,  pT  and  pD , change 
with  the  mole  fraction  of  the  solute  is  illustrated  graphically 
in  Fig.  21. 

The  above  rather  detailed  treatment  of  these  three  quantities 
has  been  deemed  advisable  because  of  the  great  confusion 
regarding  them  which  exists  in  the  literature.  The  term  osmotic 
pressure  is  loosely  used  to  designate  all  three  quantities,  fre- 
quently without  any  appreciation  of  the  difference  between 
them.  Osmotic  pressure  is  frequently  spoken  of  as  the'“  cause  of 
osmosis,”  and  attempts  are  made  to  calculate  the  diffusion  pres- 
sure from  the  freezing  point.  Neither  the  diffusion  pressure  nor 
the  thermal  pressure  can  be  calculated  from  the  freezing  point, 
as  there  is  no  known  relation  connecting  them.  The  osmotic 
pressure  can  be  so  calculated,  however,  (see  Table  XVII)  and 
if  the  concentration  of  the  solute  in  the  solution  is  so  small  that 
the  value  of  its  diffusion  pressure  and  the  value  of  the  osmotic 
pressure  are  within  say  0.1  per  cent,  of  each  other,  then  the  value 
of  the  diffusion  pressure  would  obviously  also  be  known  within 
that  limit.  The  calculation  of  diffusion  pressure  in  such  a man- 
ner, however,  obviously  involves  an  a priori  knowledge  of 


Sec.  9] 


SOLUTIONS  IV 


163 


Fig.  21. — Illustrating  the  Difference  in  the  Quantities,  Osmotic  Pres- 
sure, Thermal  Pressure,  Diffusion  Pressure  and  Perfect  Gas  Pressure  for  a 
Solution  under  Constant  Pressure  and  at  Constant  Temperature. 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


164 

the  amount  by  which  the  numerical  values  of  the  two  pressures 
differ  from  each  other  for  the  solution  in  question  and  there  is 
no  very  reliable  method  for  determining  this  except  by  means  of 
the  direct  determination  of  the  diffusion  pressure  itself,  which 
can  be  accomplished  by  means  of  rate-of-diffusion  measurements. 
The  measurement  of  the  rate  of  diffusion  of  the  solute  in  a given 
solution  is,  in  fact,  the  only  safe  and  certain  method  for  deter- 
mining its  diffusion  pressure. 


The  Boiling-Point  Laws 


10.  The  General  Boiling-point  Law. — If  to  a solution  (of  a 
non-volatile  solute)  at  its  boiling  point,  TB , we  add  dNi  moles  of 
solute,  thus  increasing  its  mole  fraction  by  d:ri  and  decreasing  that 
of  the  solvent  by  dx,  then  the  decrease,  — dp,  in  the  vapor  pres- 
sure of  the  solvent  will,  according  to  equation  (18),  be 

-dp=-p(^  (32) 


If  we  now  increase  the  vapor  pressure  to  its  original  value  by 
raising  the  temperature  of  the  solution  from  TB  to  TB+dTB, 
the  solution  will  boil  again  and  this  increase  in  vapor  pressure  as 
expressed  by  equation  (1,  XII)  can  be  placed  equal  to  the  decrease 
given  by  equation  (32).  We  thus  obtain  the  relation, 


Lv 

vqTb 


dTB=-p 


dx 

x 


(33) 


or  rearranging  and  putting  RTB  in  place  of  pv o,  we  have 


d Tb  = - 


RTb 2 
Lv 


d loge  x 


(34) 


From  equation  (10,  XII)  we  obtain  TB  = TBo+AtB  and  d TB  = 
d(A^).  Making  these  substitutions  in  equation  (34)  we  have 

d(A<B)  d logc  x (35) 

L/v 

In  order  to  integrate  this  equation  it  is  first  necessary  to 
express  Lv  as  a function  of  A tB  which  can  be  done  thermody- 
namically in  terms  of  the  specific  heats  of  the  pure  solvent  in 
the  liquid  and  vapor  states  respectively. 


Sec.  11] 


SOLUTIONS  IV 


165 


If  Lv  in  a given  case  can  be  regarded  as  practically  constant 
over  the  range  covered  by  A tB,  the  integral  would  be 


log  10  x = 


-0.4343L* 

RTbo 


A ts 
Tb 


(36) 


In  Fig.  22  a comparison  is  shown  between  the  values  of  A tB 
calculated  from  the  general  integral  of  equation  (35)  and  those 
obtained  by  direct  measurement  in  the  case  of  some  solutions 
in  benzene. 

11.  Dilute  Solutions. — With  the  aid  of  Maclaurin’s  Theorem 
the  integral  of  equation  (35)  can  be  obtained  in  a form  in  which 
the  boiling-point  raising,  A tB,  is  expressed  as  a power  series  in 
the  mole  fraction,  Xi,  of  the  solute  (cf.  equation  26)  and  if  Xi  is 
small  enough  the  terms  of  the  series  which  contain  its  higher 
powers  can  be  neglected  in  comparison  with  the  term  containing 
its  first  power.  If  this  is  done,  the  series-integral  reduces  to  the 
expression, 


_RTBo 2 _RTBo 2 N i 

L,  Xl  Lv  N+Ni 


(37) 


This  is  frequently  the  most  convenient  form  of  the  equation  to 
employ  for  dilute  solutions.  If  the  solution  is  so  dilute  that  N i 
can  be  neglected  in  comparison  with  N in  the  denominator,  we 
may  write 

Mb  = RTl  - °-  y (approx.)  (38) 

and  if  we  let  N 1 be  the  number  of  moles  of  solute  in  1000  grams 
of  solvent  this  equation  becomes 

AiB=(iooSr)^i(approx-)  (39) 

where  M is  the  molecular  weight  of  the  solvent.  The  quantity 
in  the  parenthesis  is  evidently  composed  of  constants  which  are 
characteristic  of  the  solvent  only,  and  are  quite  independent  of 
the  nature  of  the  solute.  This  parenthesis  is  called  the  molal 
boiling-point  raising  for  the  solvent  in  question  and  is  represented 
by  the  symbol,  kB.  Equation  (39)  may,  therefore,  be  written 


AtB  = JcBNi  (approx.) 


(40) 


166 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


Fig.  22. — The  Boiling  Point  Law  for  Ideal  Solutions.  The  curve  is  the 
graph  of  the  integral  of  the  theoretical  equation  (35).  The  points  repre- 
sent observed  values  of  A Ib  for  solutions  of  naphthalene  and  of  diphenyl 
in  benzene.  (Measurements  by  Washburn  and  Read,  Jour.  Amer.  Chem. 
Soc.,  1915.) 


100  X! 


Sec.  11] 


SOLUTIONS  IV 


167 


In  using  equations  (36),  (37),  and  (40)  it  should  be  remembered 
that  the  last  two  are  only  approximate  ones  and  that  except  for 
very  small  values  of  A tB  the  three  equations  will  give  somewhat 
different  results  in  a molecular  weight  calculation.  In  fact  if 
either  equation  (37)  or  equation  (40)  is  employed,  due  regard 
must  be  given  to  their  limitations  in  comparison  with  the 
accuracy  with  which  the  value  of  A tB  is  measured  and  the  desired 
accuracy  in  the  molecular  weight,  for  the  case  in  question.  For 
example,  suppose  that  in  order  to  determine  the  molecular  weight 
of  a substance  in  aqueous  solution  at  100°,  the  boiling-point 
raising  for  the  solution  is  determined;  and  suppose  that  a solution 
containing  200  grams  of  the  substance  in  1000  grams  of  water 
gives  AtB  = 1.000°,  accurate  to  0.001°.  If  we  calculate  the  mo- 
lecular weight  of  the  substance,  we  find  97.2  from  equation  (36), 

98.6  from  equation  (37),  and  102.2  from  equation  (39  or  40). 
If  an  accuracy  of  1 per  cent,  is  desired,  it  is  evidently  necessary 
to  use  equation  (36)  in  calculating  the  value  of  Mi  in  this 
instance.  The  use  of  equation  (39)  would  give  a result  in  error 
by  5 per  cent. 

If  benzene  were  the  solvent  instead  of  water  and  if  38.73  grams 
of  the  substance  dissolved  in  1000  grams  of  benzene  gave  A tB  = 
1.000°  accurate  to  0.001°,  then  the  calculated  values  of  Mi 
would  be  97.2  from  equation  (36),  98.5  from  equation  (37),  and 

101.6  from  equation  (39  or  40),  very  nearly  the  same  values  as 
in  the  case  of  water. 

Problem  9a. — Perform  the  calculation  indicated  above  for  the  same  sub- 
stance dissolved  in  alcohol,  whose  heat  of  vaporization  at  the  boiling  point, 
79.8°,  is  990  cal.  per  mole. 

In  practice  equation  (40)  is  usually  employed  in  molecular 
weight  determinations  by  the  boiling-point  method  and  the 
constant,  kB,  is  determined  by  first  finding  the  value  of  A tB  for 
a solute  of  known  molecular  weight,  and  then  solving  equation 
(40)  for  kB.  This  empirical  value  of  kB  is  then  employed  in 
calculating  the  value  of  Mi  for  a solute  of  unknown  molecular 
weight.  This  method  of  operation  usually  gives  a more  accurate 
value  of  Mi  than  is  the  case  when  kB  is  calculated  from  Lv, 
owing  to  a partial  compensation  of  errors.  In  general,  however, 
it  is  better  and  is  always  safer  to  employ  equation  (37)  or  better 
still  equation  (36)  and  to  determine  Lv,  if  necessary,  by  a boiling 


168 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


point  determination  with  one  or  more  solutes  of  known  mo- 
lecular weights. 

Problem  10. — The  heat  of  vaporization  of  benzene  at  its  B.  P.,  80.2°,  is 
94.4  cal.  per  gram.  Calculate  the  molal  B.  P.  raising  for  benzene. 

Problem  11. — A solution  of  68  grams  of  a certain  substance  in  1000  grams 
of  benzene  boils  1.00°  higher  than  pure  benzene.  What  is  the  molecular 
weight  of  the  substance? 

Problem  12. — A solution  of  66  grams  of  camphor,  CioHuO,  in  1000  grams 
of  ethyl  alcohol,  C2H60,  boils  at  79.31°.  The  pure  alcohol  boils  at  78.81°. 
Calculate  the  latent  heat  of  vaporization  of  alcohol  at  its  boiling  point. 
(Use  equation  (37)). 

Problem  13. — The  latent  heat  of  vaporization  of  water  at  100°  is  540.7 
cal.  per  gram.  Approximately  how  many  grams  of  dextrose,  C6Hi206, 
must  be  dissolved  in  100  grams  of  water  in  order  to  raise  its  boiling  point 
0.1°? 


The  Freezing  Point  and  Solubility  Laws 

12.  The  General  Freezing  Point  and  Solubility  Law. — If  we 

put  T=Tf  in  equation  (19)  and  then  combine  with  equation 
(17,  XII)  so  as  to  eliminate  dll,  we  have  the  desired  relation, 
which  is 

RTf 2 dx 
Lf  x 

From  equation  (11,  XII)  we  have  TF=  TFo—AtF  and 
— d(A^)  and  hence 

— R(TFo  —Atp)2 


d Tf  = 


(41) 
d Tf  = 


d(A£p»)  — 


d loge  x 


(42) 


It  can  be  shown  thermodynamically  that 

LF  = LFo — ASp(Atp)  (43) 

where  LFo  is  the  molal  heat  of  fusion  of  the  pure  solvent  at  its 
freezing  point  and  A SP  is  the  difference  in  the  molal  heat  capac- 
ities of  the  solvent  in  the  liquid  and  crystalline  states,  respect- 
ively. Combining  this  relation  with  equation  (42)  gives 

,,  x ~R(TFo 
‘ (AW  “ Lp,  —ASp(Atf) 
and  on  integration  we  find 


, d log*  x 


(44) 


It  logf  x = A, S'/,  log,. 


(Tp  p —AtP)  LPll—ASp(Atp)  Lf, 
T„  (TF-AtF)  +Tf„ 


(45) 


Sec.  12] 


SOLUTIONS  IV 


169 


as  the  general  equation  for  the  freezing  point  lowering  of  an  ideal 
or  a dilute  solution.  In  this  equation  x is  the  mole  fraction  of 
the  solvent  in  the  solution  and  here,  by  the  term  “solvent,”  is 
meant  that  constituent  whose  pure  crystals  are  in  equilibrium 
with  the  solution.  TFo  is  the  absolute  melting  point  of  these 
crystals  and  LFo  is  their  molal  heat  of  fusion  at  P°Fo.  It  may 
happen  in  a given  case  that  A SP  is  so  small  that  it  can  be 
neglected,  and  then  the  above  equation  reduces  to 


or 


R loge  x = 


■LFo  A tF 


log  10  x = 


Tf<>  Tf 
-0.4343LFoA^ 
RTFoTf 


(46) 

(47) 


Problem  14. — Pure  naphthalene,  Ci0  H8,  melts  at  80.09°  C.  and  its  molal 
heat  of  fusion  is  4560  calories.  At  what  temperature  will  a solution  com- 
posed of  76.9  grams  of  naphthalene  and  61.6  grams  of  diphenyl,  Ci2Hi0,  be 
in  equilibrium  with  crystals  of  pure  naphthalene?  (A Sp  is  negligible.) 
s Problem  15.— Pure  diphenyl  melts  at  68.95°  C.  and  its  molal  heat  of 
fusion  is  4020  calories.  At  what  temperature  will  a solution  composed  of 
46.1  grams  of  diphenyl  and  89.6  grams  of  naphthalene  be  in  equilibrium  with 
crystals  of  pure  diphenyl.  (A*Sp  is  not  known.) 


A solution  made  up  of  two  components,  A and  B,  has  at  least 
two  freezing  points , one  being  the  temperature  at  which  the  solu- 
tion is  in  equilibrium  with  pure  crystals  of  component  A,  and 
the  other  the  temperature  at  which  it  is  in  equilibrium  with  pure 
crystals  of  component  B.  The  freezing  point  diagram  for  such  a 
system  will  therefore  consist  of  two  curves,  one  of  which  * repre- 
sents the  temperatures  at  which  solutions  of  varying  composition 
are  in  equilibrium  with  crystals  of  component  A,  and  the  other 
of  which  represents  temperatures  at  which  the  solutions  are  in 
equilibrium  with  crystals  of  component  B.  Such  a diagram  is 
shown  in  Fig.  23,  mole  fractions  being  plotted  as  abscissae  and 
freezing  points  as  ordinates.  The  temperature  at  which  these 
two  curves  intersect  each  other  is  called  the  eutectic  point  and 
is  the  temperature  at  which  a solution  having  the  composition 
shown  by  the  abscissa  of  the  point  is  in  equilibrium  with  both 
crystalline  phases  at  the  same  time.  Evidently  there  is  only  one 
temperature  and  one  composition  of  solution  for  which  this  con- 
dition can  exist.  For  an  ideal  solution  the  two  freezing  point 


170 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIY 


curves  will  be  the  graphs  of  equation  (47)  [or  more  accurately  of 
equation  (45)]  for  the  two  components,  respectively,  and  by 
solving  these  equations  simultaneously,  the  position  of  the 
eutectic  point  can  be  readily  calculated.  A comparison  of  the 
values  thus  calculated  with  values  obtained  by  direct  measure- 
ment for  the  systems,  naphthalene-diphenyl,  diphenyl-benzene, 
and  naphthalene-benzene,  is  given  in  Table  XVIII  below  and 
the  complete  freezing-point  diagram  for  these  three  systems  is 
shown  in  Fig.  24. 

The  curves  shown  in  this  figure  are  frequently  called  solubility 
curves  as  well  as  freezing  point  curves.  The  reason  for  this  dual 


A —A  0.9  0.8  0,7  0.6  0.5  0.4  0.3  0.2  0..1  0 “A 

-C MOLE  FRACTIONS > 

Fig.  23. — Typical  Freezing  Point  Diagram  for  a two  Component  System* 

naming  is  the  following : If  we  take  any  point,  P,  on  the  naphtha- 
lene curve  (Fig.  24),  the  ordinate  of  this  point  represents  the 
temperature  at  which  a solution  having  the  composition  given 
by  the  abscissa  will  begin  to  deposit  crystals  of  pure  naphthalene 
when  it  is  cooled  down  from  some  higher  temperature.  In 
other  words  it  is  the  freezing  point  of  the  solution  with  respect  to 
naphthalene.  On  the  other  hand,  if  we  take  a quantity  of  pure 
liquid  diphenyl  and  shake  it  with  an  excess  of  naphthalene  crystals 
at  the  temperature  corresponding  to  the  ordinate  of  the  point, 
P,  the  naphthalene  will  “dissolve”  in  the  diphenyl  until  the  com- 
position of  the  resulting  solution  reaches  the  value  given  by  the 


FREEZING  POINTS 


Sec.  12] 


SOLUTIONS  IV 


171 


abscissa  of  the  point,  P.  The  quantity  of  naphthalene  which 
dissolves  under  these  conditions  is  called  the  solubility  of  naph- 
thalene in  diphenyl  at  the  temperature  in  question.  Obviously 


<— MOLE  FRACTION  —> 

Fig.  24. — Freezing  Point — Solubility  Diagram  for  the  Systems,  Naph- 
thalene-diphenyl, Napthalene-benzene  and  Diphenyl-benzene.  The 
curves  are  the  graphs  of  the  theoretical  freezing  point  equation  (47)  and 
are  named  according  to  the  substance  present  as  the  crystalline  phase. 

the  curve  shows  how  this  solubility  varies  with  the  temperature. 
Hence  the  name,  “ solubility  curve.”  Solubility  may  be  expressed 
in  any  one  of  the  various  ways  in  which  composition  of  a solu- 
tion is  expressed,  as  explained  in  Section  5 of  Chapter  XI. 

Problem  16. — Calculate  the  solubility  of  naphthalene  in  benzene  at  0°  C. 
The  data  necessary  for  the  calculation  are  given  in  Table  XVIII.  Express 
the  result  as  grams  of  naphthalene  per  hundred  grams  of  solution.  Cal- 
culate also  the  solubility  of  naphthalene  in  toluene  at  25°  in  the  same  units. 


172 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


This  problem  is  given  in  order  to  emphasize  the  fact  that,  at 
any  given  temperature,  the  solubility  (expressed  as  the  molecular 
fraction  of  the  molecular  species  in  question)  of  any  pure  crystal- 
line substance  in  a liquid  with  which  it  forms  an  ideal  solution 
depends  only  upon  the  heat  of  fusion  and  the  melting  point  of 
the  crystalline  substance  in  question  and  is  not  dependent  upon 
the  particular  liquid  in  which  it  is  dissolved. 

Table  XVIII 

Comparison  of  the  eutectic  temperatures  calculated  from  the  freez- 
ing point  law  for  an  ideal  solution  with  the  values  obtained  by  direct 
measurement. 


Measurements  by  Washburn  and  Read. 


Substance 

Melting 

point 

t°F 

Heat  of 
fusion 
per  mole 
(calories) 
Lp 

Observer 

. Calculated 
Observed 

! eutectic 
eutectic  i , 

tempera- 

tempera- 

ture 

ture  , 

0 (equation 

1 E \ 47) 

Benzene 

5.48 

2370 

Demerliac 

Benzene-naphthalene 

c6h6 

J.  Meyer 

-3.48 

-3.56 

Diphenyl 

68.95 

4020 

Washburn 

! 

Benzene-diphenyl 

c6h8-c6H6 

and  Read 

-5.8 

-6.1 

Naphthalene 

C4H4-C2- 

C4H4 

80.09 

4560 

Alluard 

N aphthalene-diphenyl 

Pickering 

39.4 

I L 

39.4 

13.  Freezing  Points  of  Dilute  Solutions. — When  the  mole 
fraction  of  the  solute  is  small  we  may  integrate  equation  (44) 
by  means  of  Maclaurin’s  Theorem  in  exactly  the  same  way  as 
in  the  case  of  the  corresponding  boiling  point  equation  and  obtain 
the  similar  equations, 


AJ  RTF * RTf.*  N 1 , . 

A<f=  Lr.  Xi=  lf.  N+m  (approx-) 

(48) 

A,  RTp*  Ni  , v 

Mf=  Ljl  n (approx.) 

(49) 

. , MRTf*  Ar  , s 

A<  = 1000  Lf.  ^(aPProx-) 

(50) 

and 

A tp  = kpN  1 

(51) 

Sec.  13] 


SOLUTIONS  IV 


173 


* 


which  correspond  exactly  to  the  boiling  point  equations  (37), 
(38),  (39)  and  (40)  and  are  applicable  under  analogous  conditions. 
The  constant,  kF,  is  called  the  molal  freezing  point  lowering 
for  the  solvent  in  question.  Equations  (49-51)  are  usually 
known  as  the  Raoult-Van’t  Hoff  equations. 

In  words  equation  (51)  states  that  as  the  solution  becomes  more 


and  more  dilute  the  molal  freezing  point  lowering, 


should 


approach  a constant,  kF,  which  is  characteristic  of  the  solvent 
and  which  can  be  calculated  from  the  relation, 

MRTFo2 


kF  = 


(52) 


1000  LFo 

For  aqueous  solutions  we  have,  M = 18,  R=  1.985  (see  II,  7), 
7V0  = 0°+273.1°,  = 273.1°,  and  LF=  18X79.60  cal.  (see  X,  6) 
and  hence  ^ = 1.86°.  Bedford4  determined  directly  the  value  of 

the  ratio,  for  a series  of  cane  sugar  solutions  of  concentra- 
tions ranging  between  0.005  and  0.04  molal  and  found  it  to  be 
constant  within  the  limits  of  experimental  error  and  equal  to 
1.86°.  (See  Fig.  27.) 

For  aqueous  solutions  of  substances  such  as  the  alcohols  and 
sugars  which  resemble  water  somewhat,  the  thermodynamic 
environment  may  be  expected  to  remain  approximately  constant 
up  to  concentrations  as  high  as  one  or  two  weight-molal,  and 
for  such  high  concentrations  the  general  freezing  point  equation 
(45)  should  always  be  employed  rather  than  any  of  its  approxi- 
mate forms  (equations  48  to  52).  With  the  aid  of  Maclaurin’s 
Theorem  this  general  equation  can  be  put  in  the  form  of  a series. 
If  this  is  done  and  the  numerical  values  of  the  constants  for  water 
are  substituted,  the  two  expressions, 


xi  = 0.009690(A  tF  - 0.00425A  &F)  (53) 

A tF  = 103.20(x1+0.428x12)  (54) 

are  obtained.  For  ideal  solutions  in  which  water  is  the  solvent 
these  expressions  are  accurate  to  0.001°  provided  A tF  is  not 
greater  than  7°. 

In  Fig.  25  is  shown  the  graph  of  these  equations  togethre  with 
the  observed  values  of  A tF  for  methyl  and  ethyl  alcohols  and  for 
cane  sugar.  The  agreement  in  the  case  of  the  alcohols  is  very 


174 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


good  but  the  observed  points  for  cane  sugar  fall  below  the  theo- 
retical curve.  If,  however,  we  assume  that  cane  sugar  exists 
in  solution  in  the  form  of  the  hydrate,  Ci2H220i]-6H20,  and 
calculate  the  mole  fractions  of  this  hydrate  by  means  of  equation 
(23)  the  points  thus  obtained  (dotted  circles  in  the  figure)  fall 


Fig.  25. — The  Freezing  Point  Law  for  Aqueous  Solutions.  The  curve 
is  the  graph  of  the  theoretical  equation,  (52).  The  points  represent  ob- 
served values  of  A tp  for  the  substances  indicated,  [Technology  Quarterly, 
21,  376  (1908)]. 

exactly  on  the  theoretical  curve,  thus  confirming  the  osmotic 
pressure  results  for  these  solutions  (see  Table  XVII).  The 
results  for  the  alcohols  indicate  that  they  are  either  not  hydrated 
in  aqueous  solution  or  else  that  the  hydrate  is  one  containing 
not  more  than  one  molecule  of  water.  The  data  do  not  warrant 
a more  exact  conclusion  than  this  owing  to  the  assumption  of 
18  as  the  molecular  weight  of  water. 


Sec.  14] 


1 SOLUTIONS  IV 


175 


Problem  17. — Calculate  the  freezing  point  of  a weight  molal  solution  of 
chloral  hydrate,  CCLCOITH/),  in  water. 

Problem  18. — From  the  data  given  in  Table  XVIII  calculate  the  volume 
of  methane,  CH4  (measured  under  standard  conditions)  which  must  be 
dissolved  in  1000  grams  of  benzene  in  order  to  lower  its  freezing  point  0.6°. 

14.  The  Solubility  Law  for  Dilute  Solutions. — In  the  case  of 
dilute  solutions  the  term  freezing-point  curve  is  usually  restricted 
to  the  curve  representing  the  temperatures  at  which  solutions  of 
varying  composition  are  in  equilibrium  with  the  pure  crystalline 
solvent;  while  the  term,  solubility  curve , is  applied  only  to  the 
curve  with  represents  the  compositions  of  solutions  which  at 
different  temperatures  are  in  equilibrium  with  the  pure  crystals 
of  the  solute.  In  the  former  case  the  heat  of  fusion  of  the  sub- 
stance present  in  the  crystalline  state  ( i.e .,  the  solvent)  is  iden- 
tical with  its  heat  of  solution  in  the  “dilute  solution,”  for  by 
definition  (XIII,  3)  this  solution  has  the  same  thermodynamic 
environment  as  that  which  prevails  in  the  pure  liquid  solvent. 
In  the  latter  case,  however,  this  condition  is  not  true  except  for 
ideal  solutions,  for  the  thermodynamic  environment  in  a dilute 
solution  is  in  general  very  different  from  that  which  prevails 
in  the  pure  liquid  solute.  Equation  (41)  applies  to  both  cases 
but  in  the  latter  case  in  place  of  LF,  the  molal  heat  of  fusion  of 
the  solute  crystals,  we  must  write  Lst.,  the  molal  heat  of  solution 
of  these  crystals  in  enough  solvent  to  form  a “dilute  solution.” 
Equation  (41)  would,  therefore,  read 


dT=  ~ 

Lsi.  xi 

(55) 

d log,  Xi\  Lsi. 

, dT  Jp  RT 

1 

(56) 

where  T is  the  temperature  at  which  the  crystals  of  the  solute 
are  in  equilibrium  with  a dilute  solution  in  which  the  mole  frac- 
tion of  the  solute  is  x\  and  LSi . is  the  heat  absorbed  when  one  mole 
of  the  solute  crystals  is  dissolved  in  enough  of  the  solvent  to 
form  a “dilute  solution.” 

If  we  assume  Lsi . constant,  the  integral  of  this  equation  is, 

, x\  0.4343LSZ.  (1  1 \ 

logl°  X\  ~ R \T  T'J 


(57) 


176  PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 
or,  since  X\  is  small, 


(approx.) 


(58) 


where  S is  the  solubility  of  the  solute  (expressed  in  any  desired 
units)  at  the  temperature,  T,  and  S'  is  its  solubility  at  the  tem- 
perature, T' . 

Problem  19. — The  solubility  of  iodine,  I2,  in  water  is  0.001341  mole  per 
liter  at  25°  and  0.004160  mole  per  liter  at  60°.  What  is  the  molal  heat  of 
solution  of  iodine  within  this  temperature  range?  What  is  the  solubility  of 
iodine  at  40°  ? 

15.  The  Application  of  the  Solution  Laws  to  the  Interpretation 
of  Processes  Occurring  within  the  Solution,  (a)  The  Dis- 
tribution Laws. — Since  all  of  the  solution  laws  discussed  in  this 
chapter  involve  the  molecular  weight  of  at  least  one  of  the  con- 
stituents, they  are  made  the  basis  for  the  determination  of  molec- 
ular weights  in  solution.  In  employing  the  laws  for  this  purpose 
the  assumptions  implied  in  their  derivation  and  the  exact  signifi- 
cance of  the  quantities  which  they  involve  must  be  kept  clearly  in 
mind.  Thus  in  using  any  one  of  the  distribution  laws  it  must 
always  be  remembered  that  these  laws  are  derived  only  for  the 
distribution  of  a given  molecular  species  and  it  is  the  mole  fraction 
or  concentration  of  the  molecular  species  in  question  which  belongs 
in  the  equation  of  the  law  and  not  the  total  mole  fraction  of  the 
solute,  if  the  latter  happens  to  be  made  up  of  more  than  one  molec- 
ular species.  For  example,  when  acetic  acid  is  in  distribution 
equilibrium  between  water  and  benzene  there  are  present  in  the 
system  the  two  molecular  species,  CH3COOH  and  (CH3COOH)2 
and  each  species  will  distribute  itself  between  the  two  phases  in 
accordance  with  the  distribution  law  and  will  have  its  own  char- 
acteristic distribution  constant.  But  if  we  were  to  substitute 
in  the  distribution  law  (equation  (15))  the  mole  fraction  of  acetic 
acid  in  each  phase,  arbitrarily  calculated  on  the  assumption  that 
it  exists  only  as  CIi3COOH-molecules  in  both  phases,  we  would 
find  that  the  “ distribution  constant”  thus  calculated  was  far 
fropi  being  constant.  In  other  words,  the  law  would  apparently 
not  be  obeyed.  The  failure  of  the  law  in  such  a case  would  be 
only  an  apparent  one,  however,  owing  to  the  fact  that  we  have 
through  ignorance  not  substituted  the  proper  values  in  the 


Sec.  15] 


SOLUTIONS  IV 


177 


equation.  We  have  an  example  of  a deviation  of  this  character 
in  Table  XV  where  the  values  of  the  calculated  distribution  con- 
stant are  seen  to  decrease  slightly  but  steadily  as  Cw  increases. 
It  is  known  from  independent  evidence,  however,  that  the  mer- 
curic bromide  is  partially  dissociated  in  the  water  layer  and  that 
the  degree  of  this  dissociation  increases  with  the  dilution  of  this 
layer,  that  is,  it  increases  as  Cw  decreases.  The  result  is  that  the 
true  value  of  Cw  for  the  HgBr2-molecules  is  in  reality  less  than 
the  total  concentration  of  mercuric  bromide  in  the  water  layer  by 
an  amount  which  is  proportionally  greater  the  smaller  Cw  and 
this  is  sufficient  to  account  for  the  slight  apparent  deviation 
from  the  law  shown  in  Table  XV. 

(b)  The  Osmotic,  Freezing  Point,  and  Boiling  Point  Laws. — 
With  reference  to  those  solution  laws  which  involve  one  of  the 
colligative  properties  of  the  “solvent”  (the  freezing  point,  boiling 
point,  and  osmotic  pressure  laws)  it  should  be  remembered  that 
in  the  equations  of  these  laws  (equations  25  to  29,  37  to  40,  and 
48  to  51))  the  quantities  aq,  Nh  and  C i,  referring  to  the  amount 
of  the  solute  molecules,  signify  (as  is  evident  from  the  derivation 
of  these  equations)  the  total  mole  fraction  or  concentration  of 
all  species  of  solute  molecules  in  the  solution  and  not  the  value  for 
some  one  of  these  molecular  species  as  was  the  case  for  the  dis- 
tribution laws.  For  example,  if  one  gram  molecular  weight  of 
acetic  acid  (CH3COOH)  were  dissolved  in  1000  grams  of  benzene, 
the  quantity  Ni  in  equation  (51)  would  be  unity  provided  all  of 
the  acetic  acid  when  in  solution  existed  in  the  form  of  CH3COOtl- 
molecules.  If,  however,  half  of  it  were  in  the  form  of 
(CH3COOH) 2- molecules,  the  solution  would  actually  contain 
0.25  gram  molecular  weight  of  (CH3COOH)2-molecules  and  0.5 
gram  molecular  weight  of  CH3COOH-molecules  or  altogether 
0.75  gram  molecular  weight  of  solute  molecules  and  this  is  the 
value  for  N_i  which  would  then  belong  in  equation  (51).  Since 
the  molecular  condition  of  the  acetic  acid  in  the  benzene  is  not 
known  in  advance  we  could  evidently  obtain  some  light  upon  this 
question  by  measuring  the  freezing  point  lowering  produced  by 
known  quantities  of  acetic  acid  and  then  calculating  the  value 
of  N i and  hence  the  molecular  weight  by  means  of  equation  (51). 

(c)  Molecular  Weights  in  Soluticn. — By  the  investigation 
of  the  behavior  of  solutions  with  reference  to  the  solution  laws 


178 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


the  following  facts  regarding  the  .molecular  weights  of  substances 
in  solution  have  been  ascertained:  (1)  Almost  all  substances 
(except  salts,  acids  and  bases)  when  dissolved  in  water,  alcohol, 
ether,  acetic  acid,  or  in  general  any  solvent  containing  oxygen 
in  its  molecule  have  either  the  same  molecular  weight  as  they  have 
in  the  gaseous  state  or  a greater  molecular  weight  due  to  the  fact 
that  they  have  united  with  some  of  the  solvent  to  form  a solvate. 
(2)  The  same  statement  is  true  of  almost  all  organic  substances 
(except  those  which  contain  an  hydroxyl  group)  when  dissolved 
in  any  solvent.  (3)  Compounds  containing  hydroxyl  (alcohols, 
acids)  when  dissolved  in  organic  liquids  whose  molecules  contain 
no  oxygen  (hydrocarbons,  halogen  and  sulphur  compounds,  etc.) 
are  associated  (III,  5),  at  least  partially,  in  the  solution.  (4) 
Salts,  acids,  and  bases  in  many  solvents,  especially  in  water,  have 
apparently  molecular  weights  stnaller  than  those  corresponding 
to  their  formulas,  that  is,  they  are  dissociated,  at  least  partially, 
into  smaller  molecules.  This  dissociation,  known  as  electrolytic 
dissociation  (I,  2g),  and  the  properties  to  which  it  gives  rise  will 
form  the  topic  of  the  next  division  of  our  subject. 

(d)  The  Interpretation  of  Experimental  Data. — Probably  the 
greatest  source  of  uncertainty  in  the  use  of  the  solution  laws  for 
interpreting  the  processes  occurring  within  a solution  lies  in;the 
difficulty  of  distinguishing  between  apparent  divergencies  from 
the  laws,  such  as  those  described  above,  and  real  divergencies 
resulting  from  the  fact  that  the  thermodynamic  environment 
within  the  solution  is  not  constant.  Frequently  divergencies 
from  both  sources  are  present  simultaneously  and  it  is  necessary 
to  use  great  caution  in  drawing  conclusions  in  such  instances  and 
to  confirm  any  conclusions  drawn,  by  evidence  from  sources  of 
an  entirely  different  character.  The  interpretation  of  scientific 
data  with  the  help  of  a set  of  derived  relationships  such  as  the 
laws  of  solutions  cannot  be  intelligently  made  unless  the  deriva- 
tion of  the  laws  and  all  of  the  assumptions  involved  therein  are 
clearly  understood  by  the  one  who  is  attempting  to  employ  them. 
The  literature  of  chemistry  unfortunately  abounds  with  examples 
illustrating  the  truth  of  this  maxim  and  it  is  for  this  reason  that 
so  much  space  and  attention  have  been  devoted  to  the  derivation 
of  the  solution  laws  in  this  chapter.  It  may  be  well  to  mention 
also  at  this  point  a not  infrequent  error  in  logic  made  by  beginners. 


Sec.  16] 


SOLUTIONS  IV 


179 


Suppose,  for  example,  that  postulating  A and  B as  true  we  can 
demonstrate  by  purely  logical  reasoning  the  truth  of  C as  a neces- 
sary consequence.  Evidently  then  if  we  are  able  to  prove  by 
experiment  that  A and  B are  true,  we  at  the  same  time  establish 
the  truth  of  C.  Suppose,  however,  that  we  are  able  to  show  by 
direct  experiment  that  C is  true.  It  by  no  means  followg^from 
this  that  A and  B are  true,  however.  Before  we  ^rejustified 
in  drawing  this  conclusion  we  must  first  demonstnjjb  rhat  A and 
B are  the  only  possible  or  reasonable  assunp^tions  from  which 
the  truth  of  C can  be  shown  to  follow.  lather  words  if  a derived 
relation,  such  as  one  of  the  solution  litv^s  for  example,  is  found 
experimentally  to  hold  true  for  J^-*given  case,  it  by  no  means 
necessarily  follows  that  the  assumptions  on  the  basis  of  which 
the  relationship  was  derived  also  hold  true  for  that  case. 

Problem  20. — Show  that  equations  (28,  29,  38, 39,  49  and  50)  are  independ- 
ent of  the  molecular  weight  of  the  solvent  and  that  hence  no  information  con- 
cerningthe  molecular  weight  of  a solvent  can  be  obtained  from  freezing  point, 
boiling  point  or  osmotic  pressure  measurements  of  dilute  solutions. 

Problem  21. — A certain  molecular  species  in  dilute  solution  is  in  distri- 
bution equilibrium  between  two  solvents.  If  equation  (9)  holds  true  for  the 
vapor  of  this  species  above  the  two  solvents  show  that  equation  (16), 
Aj 

= const.,  must  also  hold  true.  If  equation  (9)  is  not  true  but  if  instead  the 

relations,  pi  = ki^j  + k' jVi2  and  p2  = k?Nj  + k'2N?2,  hold  true  for  the 
vapor  pressures  of  the  solute  from  the  two  liquid  phases  respectively,  show 
Ki 

that  the  relation  = const.,  will  still  hold  true  within  0.1  per  cent,  for  all 

cases  in  which  (jr  — —jjyj  0.001. 

Problem  22. — Two  liquids,  A and  B,  when  shaken  together  dissolve  in 
each  other  forming  two  nonmiscible  solutions.  Show  with  the  aid  of. 
thermodynamics  (X,  10)  that  neither  substance  can  obey  the  vapor  pressure 
law  of  Ideal  Solutions  (equation  3)  in  both  solutions;  in  other  words,  that  two 
liquids  cannot  form  an  Ideal  Solution  with  each  other  unless  they  are  mis- 
cible in  all  proportions. 

Concentrated  Solutions  In  General 

16.  Solutions  of  Variable  Thermodynamic  Environment. — 

Whenever  the  thermodynamic  environment  of  a solution  varies 
with  its  composition,  the  laws  developed  in  the  preceding  sections 
of  this  chapter  are  no  longer  applicable.  The  magnitude  and 


180 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


direction  of  the  departure  from  these  limiting  laws  in  any  given 
case  depends  upon  the  natures  of  the  constituents  of  the  solution 
and  for  this  reason  a satisfactory  quantitative  theory  for  solutions 
in  general  has  not  yet  been  developed  (cf.  XII,  9)  and  owing  to 
the  nature  of  the  problem  there  seems  to  be  no  immediate  likeli- 
hood that  such  a theory  will  be  formulated.  In  working,  there- 
fore, with  solutions  in  which  the  theimodynamic  environment 
cannot  be  regarded  as  constant  (and  this  is  the  case  with  most 
solutions),  each  solution  must  be  treated  as  a problem  by  itself 
and  the  quantitative  relation  connecting  some  one  of  its  colligative 
properties  (the  vapor  pressure,  for  example)  with  the  composition 


Fig.  26. 


must  be  determined  by  direct  experiment.  The  other  relations 
can  then  be  derived  with  the  aid  of  thermodynamics.  Although 
a quantitative  theory  for  solutions  in  general  is  lacking,  there  are 
a few  useful  principles  of  a qualitative  nature  which  are  applic- 
able to  all  solutions  and  which  are  of  considerable  practical  assist- 
ance in  the  intelligent  carrying  out  of  some  of  the  operations  to 
which  solutions  are  subjected  in  the  laboratory  and  the  indus- 
tries. We  shall  discuss  two  of  these  briefly. 

17.  The  Theory  of  Distillation.’ — The  boiling  point-composition 
curves  for  homogenous  liquid  mixtures  may  be  conveniently 


Sec.  18] 


SOLUTIONS  IV 


181 


divided  into  three  types  which  are  illustrated  in  Fig.  26.  In 
type  I the  boiling  point  rises  continuously  from  the  boiling  point 
of  one  pure  constituent  to  the  boiling  point  of  the  other.  The 
curve  for  ideal  solutions  is  of  this  type.  Type  II  exhibits  a 
maximum  and  type  III  a minimum  boiling  point  at  a certain 
composition. 

If  a homogeneous  liquid  mixture  of  two  constituents  be  sub- 
jected to  fractional  distillation  at  constant  pressure,  it  is  obvious 
that  as  the  distillation  progresses  the  moie  volatile  portion  of 
the  mixture  will  collect  in  the  distillate  while  the  less  volatile  por- 
tion will  remain  in  the  residue.  Or  in  other  words  the  composition 
of  the  distillate,  continuously  approaches  that  of  the  liquid  having 
the  minimum  boiling  point,  while  the  composition  of  the  residue 
continuously  approaches  that  of  the  liquid  having  the  maximum 
boiling  point.  This  is  the  principle  at  the  basis  of  the  theory 
of  fractional  distillation.  Its  application  to  specific  cases  will 
be  understood  by  the  solution  of  the  following  problems. 

Problem  23. — Each  of  the  four  solutions  having  the  compositions  indicated 
by  the  dotted  lines  in  Fig.  26  is  separated  as  completely  as  possible  into  two 
portions  by  fractional  distillation  at  constant  pressure.  What  will  be  the 
composition  of  each  portion  assuming  that  the  boiling  point  curve  is  (1) 
of  type  I,  (2)  of  type  II  and  (3)  of  type  III? 

Problem  24. — In  making  up  a standard  solution  of  hydrochloric  acid  it 
is  sometimes  convenient  to  remember  that  if  any  hydrochloric  solution  of 
unknown  strength  be  taken  and  boiled  in  the  open  air  (760  mm.)  for  a suffi- 
cient length  of  time,  it  will  attain  the  composition  20.2  per  cent.  HC1, 
boiling  at  110°,  and  may  then  be  diluted  to  the  desired  strength.  Draw  a 
diagram  showing  the  character  of  the  boiling  point  curve  for  HC1  solutions 
in  water.  See  Table  XII  for  the  adidtional  datum  required. 

18.  Distillation  of  Nonmiscible  Liquids. — When  liquids  which 
are  mutually  insoluble  (or  practically  so)  in  oner«another  are  heated 
together  the  mixture  will  boil  when  the  sum  of  the  separate  vapor 
pressures  of  the  components  reaches  a value  equal  to  the  external 
pressure;  on  the  mixture.  This  temperature  can  be  readily 
determined  if  the  vapor  pressure  curves  of  the  component  liquids 
are  known.  It  will  necessarily  be  lower  than  any  of  the  boiling 
points  of  the  components  and  may  be  far  lower  than  some  of 
them.  This  makes  it  possible  to  effect  the  distillation  of  a high 
boiling  liquid  at  a temperature  much  below  its  boiling  point  with- 


182 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIV 


out  having  recourse  to  vacuum  distillation  and  is  the  principle 
at  the  basis  of  the  process  of  distillation  by  steam  which  is  fre- 
quently employed  in  the  purification  by  distillation  of  such  liquids 
as  aniline,  nitrobenzene  and  oils,  which  are  insoluble  in  water. 
The  composition  of  the  distillate  from  such  a distillation  can  be 
calculated  by  means  of  Dalton’s  law  of  partial  pressures  (II,  6) 
if  the  vapor  pressures  of  the  pure  components  at  the  boiling  point 
of  the  mixture  are  known. 

Problem  25. — A current  of  steam  is  passed  at  atmospheric  pressure 
through  a mixture  of  water  and  chlorbenzene  (CeHsCl).  From  the  data 
recorded  in  the  Landolt-Bornstein  tables,  determine  the  boiling  point  of 
the  mixture  and  the  per  cent,  by  weight  of  chlorbenzene  in  the  distillate. 

19.  The  Theory  of  Fractional  Crystallization. — When  a solu- 
tion is  fractionated  by  distillation,  the  theory  of  the  process  is  much 
simplified  by  the  fact  that  only  one  gaseous  phase  is  possible 
(I,  8)  in  the  systepi.  When  the  fractionation  is  carried  out  by 
crystallization,  however,  our  problem  may  become  a very  com- 
plicated one  owing  to  the  large  number  of  different  crystal  phases 
which  are  possible.  The  general  principle  underlying  the  theory 
of  fractional  crystallization  may  be  stated  as  follows:  When  a 
homogeneous  liquid  mixture  of  two  constituents  is  subjected  to  frac- 
tional crystallization  the  composition  of  the  mother  liquor  continuously 
approaches  the  liquid  having  the  lowest  crystallization  temperature. 
This  liquid  is  usually  a eutectic  mixture  of  some  kind  although  it 
may  be  one  of  the  pure  constituents.  The  crystals  which  separate 
during  the  fractionation  may  be  either  (1)  one  of  the  pure  constit- 
uents, (2)  a compound  between  the  two,  (3)  mixed  crystals  con- 
taining both  constituents,  or  (4)  mixed  crystals  containing  one  of 
the  constituents  and  a compound  between  the  two.  We  shall  not 
here  enter  upon  the  detailed  discussion  of  these  special  cases 
as  they  are  more  conveniently  taken  up  in  connection  with  the 
discussion  of  the  Phase  Rule  and  its  applications. 

REVIEW  PROBLEMS 

Problem  26. — The  vapor  pressure  of  ether  (C4H10O)  at  13°  is  33.18  cm. 
This  is  lowered  by  0.80  cm.  by  dissolving  something  in  the  ether.  The 
specific  gravity  of  ether  is  0.721.  Calculate  the  osmotic  pressure  in  atmos- 
pheres of  this  solution.  (See  prob.  1,  Chap.  XII.) 

Problem  27. — The  value  0.0198  for  the  relative  vapor  pressure  lowering 
of  a weight  molal  solution  of  cane  sugar  in  water  at  25°  has  been  found  by 


Sec.  19] 


SOLUTIONS  IV 


183 


direct  experiment.  Calculate  the  osmotic  pressure  of  the  solution.  (An- 
swer: 27.1  atm.  Compare  with  the  observed  osmotic  pressure  given  in 
Table  XVI.)  (See  prob.  1,  Chap.  XII.) 

Problem  28. — At  30°  the  vapor  pressure  of  pure  ether  (C4H10O)  is  64 
cm.  and  that  of  pure  water  is  3.1  cm.  When  the  two  liquids  are  shaken 
together  at  30°,  1 gram  of  water  is  dissolved  by  73  grams  of  ether  and  1 
gram  of  ether  by  18.8  grams  of  water.  Calculate  the  total  vapor  pressure 
above  the  mixture. 

Problem  29. — Five  liters  of  air  measured  at  20°  and  1 atm.  are  passed 
through  a saturater  containing  liquid  ether  at  20°  and  the  air  saturated 
with  ether  vapor  issues  from  the  saturater  under  a total  pressure  of  1 
atmosphere.  The  vapor  pressure  of  ether  at  20°  is  44.2  cm.  How  many 
grams  of  ether  will  be  vaporized? 

Problem  30. — In  problem  10  if  the  CC14  solution  is  shaken  with  9 succes- 
sive portions  of  water  of  3 liters  each,  how  many  grams  of  iodine  will  be  left 
in  the  CC14  layer? 

Problem  31. — The  latent  heat  of  vaporization  of  cyclohexane  is  89  cal. 
per  gram  at  its  boiling  point,  81.5°.  How  many  grams  of  anthracene  (C4Hio) 
must  be  dissolved  in  200  grams  of  cyclohexane  in  order  to  raise  its  boiling 
point  0. 1°? 

Problem  32. — At  what  temperature  will  a solution  of  1 gram  of  toluene 
(C6H5CH3)  in  1000  grams  of  benzene  be  in  equilibrium  with  crystals  of 
benzene?  See  Table  XVIII  for  additional  data. 

REFERENCES 

3 Books:  Osmotic  Pressure.  Findlay,  1913. 

4 Journal  Articles:  Bedford,  Proc.  Roy.  Soc.  Lon.  83  A,  459  (1910). 


t 


> 


INTRODUCTION 


TO  THE 

PRINCIPLES 

OF 

PHYSICAL  CHEMISTRY 


PART  II.  THE  IONIC  THEORY  AND 
CHEMICAL  EQUILIBRIUM 


BY 

EDWARD  W.  WASHBURN 

PROFESSOR  OF  PHYSICAL  CHEMISTRY  IN  THE 
UNIVERSITY  OF  ILLINOIS 


JOHN  W.  DAVIS 

APR  19  1915 


Printed  for  the  Use  of  Students  in  the 
University  of  Illinois.  Not  Published 


. Of  lyr 

' OF  H I 


McGRAW-HILL  BOOK  COMPANY,  Inc. 
239  WEST  39TH  STREET,  NEW  YORK 
6 BOUVERIE  STREET,  LONDON,  E.  C. 

1915 


Copyright,  1915,  by  the 
McGraw-Hill  Book  Company,  Inc. 


PRINCIPLES  OF 
PHYSICAL  CHEMISTRY 


c 


CHAPTER  XV 

THE  COLLIGATIVE  PROPERTIES  OF  SOLUTIONS  OF 
ELECTROLYTES 

1.  The  Molal  Freezing  Point  Lowering. — We  have  already 

A tp 

learned  (XIV,  13)  that  the  molal  freezing  point  lowering(~  = kF) 

iVi 

for  any  solvent  should  theoretically  reach  a constant  limiting 
value  in  dilute  solution  and  that  for  aqueous  solutions  this 
limiting  value  should  be  1.86°.  In  the  case  of  all  aqueous  solu- 
tions which  do  not  conduct  electricity  this  theoretical  con- 
clusion has  been  completely  and  quantitatively  confirmed  by 
direct  experiment,  not  a single  exception  to  it  having  ever  been 
discovered.  But  in  the  case  of  all  aqueous  solutions  which  do 
conduct  electricity  this  conclusion  is  apparently  never  confirmed, 
since  such  solutions  without  exception  give  decidedly  larger 
values  than  1.86°  for  the  molal  freezing  point  lowering  at  in- 
finite dilution.  These  facts  are  illustrated  graphically  in  Fig. 
27.  In  the  case  of  sugar  solutions  the  value  of  kF  is  constant 
and  equal  to  1.86°  for  all  values  of  Ni  below  0.01.  With  salts 
of  the  types  of  KC1  and  MgSOi  the  values  of  kF  are  all  larger 
than  those  for  sugar  and  gradually  approach  the  limiting  value, 
3.72°  ( = 2X1.86°)  as  N_ i decreases.  The  values  for  BaCE 
are  still  larger  and  approach  a limit  which  is  evidently  close  to 
5.58°  (=3X1.86°)  while  in  the  case  of  K3FeCy6  the  limiting 
value  must  evidently  be  close  to  7.44°  ( = 4X1.86°).  These 
figures  are  typical  of  those  obtained  with  all  solutions  of  sub- 
stances which  conduct  electricity.  The  limiting  value  of  kF 
instead  of  being  1.86°  is  invariably  some  small  exact  multiple 
of  this  number.  In  other  words  the  freezing  point  lowering 
is  always  much  larger  than  the  value  corresponding  to  the  mo- 
lecular weight  of  the  solute. 

2.  The  Other  Colligative  Properties. — From  the  results  ob- 
tained for  the  freezing  point  lowerings  of  aqueous  solutions  of 

184 


Sec.  3] 


SOLUTIONS  OF  ELECTROLYTES 


185 


substances  which  conduct  electricity  it  necessarily  follows 
(see  XII,  8)  that  the  other  colligative  properties  of  these  solu- 
tions, the  osmotic  pressure,  the  boiling  point  raising  and  the 
vapor  pressure  lowering,  must  likewise  be  abnormally  large  and 
this  is  completely  confirmed  by  the  results  of  experiment.  Thus 
the  osmotic  pressure  of  a 0.5  weight  molal  solution  of  KC1  at 


Fig.  27. — Variation  of  molal  freezing  point  lowering  with  concentration 
for  aqueous  solutions.  The  centers  of  the  circles  represent  Bedford’s  meas- 
urements. The  crosses  are  measurements  by  Adams  (Jour.  Amer.  Chem. 
Soc.,  Mar.,  1915).  Bedford  has  not  published  his  individual  values,  for  sugar, 
merely  stating  that  they  were  constant  for  the  above  range.  The  crosses 
on  the  lower  curve  are  Adams’  values  for  mannite. 

25°  has  been  found  to  be  larger  than  20  atmospheres  and  its 
relative  vapor  pressure  lowering  is  0.014.  The  corresponding 
values  for  substances  which  do  not  conduct  electricity  are  12 
atmospheres  and  0.09. 

These  facts  and  a great  many  others  concerning  the  behavior 
of  solutions  which  conduct  electricity  find  their  only  successful 
interpretation  in  the  Ionic  Theory  which  was  first  proposed  by 


186 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XV 


Arrhenius  in  1887.  In  the  following  pages  the  Ionic  Theory  will, 
therefore,  be  employed  as  the  framework  for  the  systematic  devel- 
opment and  presentation  of  our  present  knowledge  of  solu- 
tions of  electrolytes. 

3.  The  Ionic  Theory,  (a)  Electrolytic  Ionization. — According 
to  the  Ionic  Theory  when  a salt,  such  as  KC1  for  example,  is 
dissolved  in  water,  part  of  its  molecules  split  up  or  dissociate  in 
such  a way  that  the  valence  electron  (I,  2g)  of  the  potassium 
remains  attached  to  the  chlorine  atom  which  thus  becomes 
charged  with  one  unit  of  negative  electricity  while  the  potassium 
atom  is  left  with  a charge  of  one  unit  of  positive  electricity. 
Each  of  these  charged  atoms  attaches  to  itself  a certain  number  of 
water  molecules  and  the  two  electrically  charged  complexes  thus 
formed  are  termed  ions  (I,  2g).  It  is  the  presence  of  these  ions 
which  gives  to  the  solution  its  power  to  conduct  electricity  and 
any  substance  which  is  capable  of  ionizing  in  this  way  is  called 
an  electrolyte.  Positive  ions  are  called  cations  and  negative 
ions,  anions.  The  dissociation  or  ionization  of  the  potassium 
chloride  may  be  represented  symbolically  by  the  following 
reaction: 

K Cl  (solid)  -\-aq  = KC1- aq  = KC1(H20)P  = 

[K(H20)J++[C1(H20)J-  (1) 

The  symbol  aq  attached  to  the  formula  of  a substance  means 
simply  that  the  substance  is  dissolved  in  a large  amount  of 
water.  The  groups  [(K(H20)m]+  and  [Cl(H20)n]_  are  the  two 
ions.  The  numerical  values  of  m and  n in  these  expressions 
are  not  known  and  it  is  usually  customary  to  omit  the  attached 
water  in  writing  the  symbol  for  an  ion.  Thus  the  ionization 
reaction  for  KC1  is  usually  abbreviated  thus, 

KC1  <=±  K+  + Cl-  (2) 

The  character,  <=±,  indicates  that  there  is  a chemical  equili- 
brium (I,  9)  between  the  two  ions  and  the  un-ionized  molecules, 
that  is,  the  ionization  of  the  KC1  at  finite  concentrations  is  not 
complete,  a certain  number  of  un-ionized  molecules  always 
being  present  in  the  solution.  As  the  solution  is  diluted  more 
and  more,  however,  the  number  of  un-ionized  molecules  con- 
tinually decreases  and  finally  at  infinite  dilution  the  KC1  is 


Sec.  3] 


SOLUTIONS  OF  ELECTROLYTES 


187 


completely  ionized  and  for  every  mole  of  KC1  which  was  originally 
dissolved  there  are  actually  present  in  the  solution  two  moles  of 
solute  molecules,  the  two  ions.  The  molal  freezing  point  lower- 
ing of  this  solution  should,  therefore,  be  just  twice  as  great  as  the 
value  for  a substance  like  sugar  whose  molecules  do  not  break  up 
in  solution.  The  same  is  true  of  MgS04  which  ionizes  thus, 

MgS04  = Mg+++S04—  (3) 

the  two  ions  in  this  instance  being  each  charged  with  two  units  of 
electricity,  since  the  sulphate  ion  retains  both  of  the  valence 
electrons  of  the  magnesium  atom.  Barium  chloride  ionizes 
thus 

BaCl2  = Ba+++2Cl--  (4) 

giving  one  doubly  charged  barium  ion  and  two  singly  charged 
chloride  ions.  One  molecule  of  BaCl2  when  it  ionizes  gives, 
therefore,  three  solute  molecules  and  its  molal  freezing  point 
lowering  at  infinite  dilution  ought  to  be  just  three  times  1.86°  or 
5.58°.  Similarly  K3FeCy6  ionizes  thus 

K3F  eCy6  = 3K+ + F eCy  6 (5) 

to  give  three  singly  charged  potassium  ions  and  one  triply  charged 
ferricyanide  ion.  Its  molal  freezing  point  lowering  at  infinite 
dilution  ought  therefore  according  to  the  Ionic  Theory  to  be 
exactly  four  times  1.86°  or  7.44°;  all  of  which  accords  com- 
pletely with  the  experimental  results  displayed  in  Fig.  27. 

(b)  Nomenclature  and  Types  of  Electrolytes. — Ions  are 
named  from  the  atom  or  group  of  which  they  are  formed.  Thus 
all  potassium  salts  give  potassium  ion  K+  when  they  ionize,  all 
ammonium  salts  give  ammonium  ion  NH4+,  all  chlorides  in- 
cluding hydrochloric  acid  give  chloride  ion  Cl-,  all  acids  give 
hydrogen  ion  H+,  and  all  hydroxides  give  hydroxyl  ion  OH-. 

Electrolytes  are  classified  according  to  the  character  of  their 
ions  as  uni-univalent  (KC1,  HC1,  NH4N03),  which  give  two 
singly  charged  ions;  uni-bivalent  (BaCl2,  H2S04,  Ca(OH)2), 
which  give  two  singly  charged  and  one  doubly  charged  ion;  bi- 
bivalent (MgS04,  BaC204),  which  give  two  doubly  charged  ions; 
uni-trivalent  (FeCl3,  Na3POi,  K3FeCy6),  which  give  three 
singly  charged  and  one  triply  charged  ion;  bi-trivalent  (Fe2(S04)3, 


188 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XV 


Ba3(P04)2)  which  give  two  triply  charged  and  three  doubly 
charged  ions;  tri-trivalent  (FeP04),  which  give  two  triply  charged 
ions;  etc. 

Problem  1. — Write  the  ionization  reactions  of  the  above  electrolytes. 
Give  also  examples  of  still  higher  types  and  write  their  ionization  reactions. 
Name  the  ions  in  each  instance. 

4.  Degree  of  Ionization. — If  N i formula  weights  of  an  electro- 
lyte be  dissolved  in  1000  grams  of  water,  a certain  fraction  of  its 
molecules  will  ionize  and  this  fraction  is  called  the  degree  of 
ionization,  a,  of  the  electrolyte  at  this  concentration.  For  a uni- 
univalent electrolyte  the  concentration  of  each  species  of  ion  in 
the  solution  will  evidently  be  aNi  and  that  of  the  un-ionized 
molecules  (1  — a)Ni}  moles  per  1000  grams  of  water.  The 
total  concentration  of  solute  molecules  in  this  solution  will,  there- 
fore, amount  to  2<xVi  + (l  — a)N_i  or  (1  + a)N]  moles  per 
1000  grams  of  water.  In  other  words  the  number  of  formula 
weights  of  a uni-univalent  electrolyte  dissolved  in  a given  quantity 
of  water  must  be  multiplied  by  (1  + a)  in  order  to  obtain  the 
actual  number  of  moles  of  solute  in  the  resulting  solution. 

In  order  to  calculate  the  value  of  a for  a given  case  we  might 
multiply  the  quantity  Ah  in  any  one  of  the  solution  laws  by  1 + a 
and  solve  the  resulting  equation  for  a.  In  this  way  we  would, 
for  example,  obtain  from  the  freezing-point  law  for  aqueous 
solution  (equation  53,  XIV)  the  relation 

*i =1^^=+  = 0.009690  (AV-0.00425  At/)  (6) 

from  which  we  could  evidently  calculate  a value  for  a from  the  ob- 
served freezing  point  lowering  of  the  solution.  The  calcula- 
tion of  degree  of  ionization  in  this  manner,  however,  assumes 
(1)  that  the  solute  is  not  hydrated  in  solution,  and  (2)  that  the 
thermodynamic  environment  in  the  solution  is  the  same  as  that 
which  prevails  in  pure  water,  for  these  are  the  assumptions  upon 
which  equation  (53,  XIV)  was  derived.  Neither  of  these  as- 
sumptions is  true,  however.  The  ions  and  in  many  cases  the 
un-ionized  molecules  are  known  to  be  hydrated  in  solution  and 
the  whole  behavior  of  solutions  of  electrolytes  indicates  that  the 
thermodynamic  environment  even  at  fairly  small  ion  concentra- 


Sec.  4] 


SOLUTIONS  OF  ELECTROLYTES 


189 


tions  (Cf.  XIII,  3)  may  be  quite  appreciably  different  from  that 
which  prevails  in  pure  water. 

The  use  in  this  manner,  of  any  of  the  Laws  of  Solutions  of 
Constant  Thermodynamic  Environment  in  order  to  calculate 
the  degree  of  ionization  of  an  electrolyte  has,  therefore,  no  theo- 
retical justification.  These  laws  have  nevertheless  been  exten- 
sively employed  for  this  purpose  and  when  the  values  of  a 
calculated  in  this  way  are  compared  with  those  obtained  by  the 
conductivity  method  (XVII,  2b)  the  agreement  in  many  cases 
is  so  close  as  to  make  it  probable  that  the  two  erroneous  assump- 
tions mentioned  above  happen  to  produce  errors  which  are  op- 
posite in  direction  and  which,  therefore,  partially  compensate 
each  other.  Noyes  and  Falk  who  have  studied  a large  number 
of  electrolytes  in  this  way  find  that  the  degree  of  ionization  of 
uni-univalent  strong  electrolytes  in  aqueous  solution  at  concen- 
trations not  exceeding  0.1  molal  can  as  a matter  of  fact  be  calcu- 
lated from  the  freezing  point  lowering  of  the  solution  with  an 
accuracy  of  from  1 to  4 per  cent,  in  most  cases.  (Cf.  Table  XIV 
and  XVII,  4.)  We  may,  therefore,  employ  the  solution  laws  in 
this  way  in  order  to  compute  approximate  values  for  the  degree 
of  ionization  of  uni-univalent  strong  electrolytes,  remembering 
that  in  so  employing  them  we  are  treating  them  merely  as 
empirical  equations.  For  this  purpose  equation  (6)  may  be 
rearranged  algebraically  into  the  more  convenient  form, 

(l+a)iVi=~g(l+0.0055A«i?)  (7) 

Problem  2. — The  degrees  of  ionization  of  KC1,  CSNO3,  and  Li  Cl  in  0.1 
weight  formal  solution  at  0°  as  measured  by  the  conductance  method  are 
0.88,  0.81,  and  0.86  respectively.  Calculate  approximate  values  for  the 
freezing  points  of  these  solutions.  (Observed  values  are  —0.345°,  —0.325°,. 
and  —0.351°  respectively.) 

Problem  3. — The  freezing  point  of  a 0.1  weight  formal  solution  of  NaBrC>3 
is  —0.342°.  Calculate  a. 

Problem  4. — By  direct  measurement  it  has  been  found  that  at  25°  a 
0.5  weight  formal  solution  of  KC1  has  the  same  vapor  pressure  as  a 0.91 
weight  formal  solution  of  mannite  (CeHuOe)  which  is  a non-electrolyte. 
From  these  two  data  estimate  approximately  the  extent  to  which  the  KC1 
is  ionized  in  the  solution. 

Electrolytes  which  are  ionized  to  a large  degree  (50  per  cent, 
or  more  in  0.1  normal  solution)  are  called  “strong  electrolytes.” 


190 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XV 


They  include  nearly  all  salts  and  all  of  the  so-called  strong  acids 
and  alkalies.  Electrolytes  which  are  only  slightly  ionized  are 
called  “weak  electrolytes.” 

5.  Complex  Formation  in  Solution. — Many  electrolytes  in 
solution  possess  the  property  of  uniting  with  other  electrolytes 
or  with  non-electrolytes  to  form  complex  compounds  which 
ionize  giving  complex  ions.  Thus  when  KCN  is  added  to  a solu- 
tion of  AgNC>3  a reaction  occurs  which  may  be  expressed  stoichio- 
metrically  as  follows: 

2KCN+AgN08  = KN03  + KAg(CN)2  (8) 

The  complex  salt  KAg(CN)2  ionizes  thus 

KAg(CN)2  = K++ Ag(CN)r  (9) 

giving  the  very  stable  complex  ion,  Ag(CN)2“ 

Problem  5. — The  freezing  point  of  a 0.1  weight  formal  solution  oi  KNO3 
is  —0.33°.  In  an  amount  of  this  containing  1000  grams  of  water,  0.1 
formula  weight  of  NH3  is  dissolved  and  the  freezing  point  then  drops  to 
— 0.51°.  A second  0.1  formula  weight  of  NH3  depresses  it  to  —0.69° 
and  a third  to  —0.9°.  If  the  same  experiments  be  carried  out  with  0.1 
weight  formal  AgN03  instead  of  KNO3,  it  is  found  that  the  freezing  point 
( — 0.33°)  ol  the  AgNC>3  solution  is  not  affected  by  the  first  two  additions  of 
NH3  but  that  the  third  addition  depresses  it  to  —0.51°.  The  electrical 
conducting  powers  of  the  AgN03  and  KNO3  solutions  are  substantially 
unchanged  by  the  additions  of  the  NH3.  What  conclusions  can  you  draw 
from  these  experiments  as  to  the  nature  of  the  complex  formed? 


CHAPTER  XVI 


THE  CONDUCTION  OF  ELECTRICITY 

1.  Classes  of  Electrical  Conductors. — Solid  and  liquid  con- 
ductors of  electricity  may  be  divided  into  two  classes:  (1)  Purely 
metallic  conductors  in  which  the  passage  of  the  current  is  not  ac- 
companied by  a transfer  of  matter  and  produces  no  effect  other 
than  an  increase  in  the  temperature  of  the  conductor;  and  (2) 
purely  electrolytic  conductors  in  which  the  passage  of  the  cur- 
rent is  accompanied  by  a simultaneous  transfer  of  matter  in 
both  directions  through  the  conductor.  Aqueous  solutions  of 
electrolytes  and  fused  salts  are  good  examples  of  electrolytic 
conductors. 

Some  interesting  conductors  which  occupy  an  intermediate 
position  between  purely  metallic  and  purely  electrolytic  con- 
ductors are  also  known.  In  these  cases  part  of  the  electricity 
which  flows  through  the  conductor  carries  no  matter  with  it 
while  the  remainder  does.  The  part  of  the  electricity  which 
passes  through  this  intermediate  class  of  conductors  in  the  first 
way,  that  is,  without  being  accompanied  by  a transfer  of  matter, 
may,  by  gradually  altering  the  composition  of  the  conductor, 
be  varied  gradually  and  continuously  from  100  per  cent,  (purely 
metallic  conduction)  to  zero  per  cent,  (purely  electrolytic  con- 
duction). Our  two  classes  of  conductors  would  seem,  therefore, 
to  constitute  simply  two  limiting  cases.  As  a matter  of  fact,  how- 
ever, most  conductors  seem  to  belong  exclusively  to  one  or  the 
other  of  these  limiting  cases,  those  of  the  intermediate  or  mixed 
character,  of  which  solutions  of  sodium  in  liquid  ammonia  con- 
stitute an  example,  being  comparatively  rare. 

If  a direct  current  is  caused  to  pass  through  an  electrolytic 
conductor  between  two  pieces  of  metal,  called  the  electrodes, 
the  electrode  at  which  the  current  enters  the  electrolytic  conductor 
is  called  the  anode  and  the  other  electrode  is  called  the  cathode. 
The  passage  of  a current  in  this  way  through  a purely  electrolytic 

191 


192 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 

conductor  is  called  electrolysis  and  is  always  accompanied  by 
chemical  changes  of  some  kind  at  both  electrodes. 

2.  The  Mechanism  of  Metallic  Conductance. — In  our  first 
chapter  (I,  2g)  mention  was  made  of  the  fact  that  in  metallic 
substances  some  of  the  electrons  are  able  to  move  about  from 
atom  to  atom  within  the  body  of  the  metal  with  comparative 
ease.  When  a difference  of  potential  is  applied  to  the  ends  of  a 
piece  of  metal  a stream  of  these  electrons  through  the  metal  is 
immediately  set  up.  This  stream  of  electrons  constitutes  the 
electric  current  in  the  metal  and  the  collision  of  these  electrons 
with  the  atoms  of  the  metal  produces  the  temperature  increase 
which  always  accompanies  the  passage  of  an  electric  current 
through  a metal. 

A metal  carrying  a current  is  surrounded  bjr  a magnetic  field, 
and  the  direction  and  nature  of  this  field  would  the  same 
whether  the  current  in  the  metal  was  caused  by  the  motion  of 
positive  electricity  in  one  direction  or  .by  the  motion  of  negative 
electricity  in  the  opposite  direction  and  before  the  nature  of 
metallic  conduction  was  understood,  the  “direction  of  the 
current”  in  a metal  was  arbitrarily  defined  as  the  direction  in 
which  positive  electricity  would  have  to  flow  in  order  to  produce 
the  observed  magnetic  effects.  We  now  know,  however,  that  the 
current  is  actually  due  to  the  motion  of  negative  electricity,  the 
electrons,  so  that  we  find  ourselves  in  the  rather  peculiar  position 
of  defining  the  “ direction  of  the  current”  in  a metal  as  the 
direction  opposite  to  that  in  which  the  electricity  actually  flows. 

3.  The  Mechanism  of  Electrolytic  Conductance. — When  a 
difference  of  potential  sufficient  to  cause  a steady  current,  is 
applied  to  two  electrodes  dipping  into  a solution  containing  dis- 
solved electrolytes,  all  of  the  anions  in  the  solution  immediately 
start  to  move  toward  the  anode  or  positive  electrode  while  all  of 
the  cations  start  moving  toward  the  cathode  or  negative  electrode. 
The  motion  of  these  ions  constitutes  the  current  of  electricity 
through  the  solution  and  the  direction  of  the  current  in  the  solu- 
tion is  arbitrarily  defined  as  the  direction  in  which  the  cations 
move.  The  quantity  of  electricity  which  is  transported  through 
the  solution  in  this  way  is  divided  among  the  different  species 
of  ions  in  the  solution  in  proportion  to  their  numbers,  the  charges 
which  they  carry,  and  the  velocities  with  which  they  move.  The 


Sec.  4] 


THE  CONDUCTION  OF  ELECTRICITY 


193 


product  of  the  number  of  charges  on  a given  species  of  ion  into 
the  number  of  these  ions  and  into  the  velocity  with  which  they 
move  through  the  solution  under  a unit  potential  gradient  is 
evidently  a measure  of  the  current  carrying  capacity  of  this 
species  of  ion  in  the  solution. 

4.  The  Mechanism  of  the  Passage  of  Electricity  between  a 
Metallic  and  an  Electrolytic  Conductor,  (a)  Electrochemical 
Reactions. — Through  any  cross  section  of  a solution  containing 
electrolytes  the  current  is  carried  by  the  ions  of  these  electro- 
lytes and  every  species  of  ion  in  the  solution , no  matter  how  great 
a number  of  different  species  there  may  be,  helps  in  carrying  the 
current,  each  species  in  proportion  to  its  current-carrying  ca- 
pacity (XVI,  3).  In  conducting  the  current  from  the  solution 
to  the  electrode  or  vice  versa , however,  only  a few,  in  many  cases 
only  one  ion  species  is  involved,  namely,  that  species  which  most 
easily  takes  on  or  gives  up  an  electron  under  the  conditions. 
Since  the  current  through  the  metallic  portion  of  the  circuit 
consists  only  of  moving  electrons  it  is  clear  that  an  exchange  of 
electrons  between  electrode  and  solution  must  take  place  in 
some  manner  at  the  electrode  surface.  This  exchange  of  electrons 
constitutes  an  electrochemical  reaction,  of  which  there  are  very 
many  varieties.  We  shall  consider  a few  such  reactions. 

Case  1. — Let  Fig.  28  represent  an  electrolytic  cell  composed  of 
two  silver  electrodes  dipping  into  a solution  of  silver  nitrate. 
The  cathode  receives  a negative  charge  from  the  dynamo,  that 
is,  a stream  of  electrons  flows  through  the  wire  from  the  dynamo 
terminal  to  the  cathode,  and  the  negative  potential  of  the 
cathode  relative  to  the  solution,  therefore,  rises.  If  this  current 
is  to  continue  flowing,  these  excess  electrons  on  the  cathode  must 
be  removed  in  some  way  by  entering  into  an  electrochemical 
reaction  with  some  species  of  ion.  The  reaction  in  this  instance 
consists  in  a union  of  the  electrons  from  the  wire  with  the  silver 
ions  in  the  solution,  which  are  thereby  reduced  to  ordinary 
silver  atoms  and  deposited  or  plated  out  upon  the  electrode.  The 
reaction  may  be  written  thus, 

Ag++(  — ) = Ag  (1) 

At  the  anode  the  silver  atoms  on  the  surface  of  the  silver  electrode 
give  up  their  electrons  and  enter  the  solution  as  silver  ions,  while 


194 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


the  electrons  thus  set  free  flow  around  through  the  wire  to  the 
positive  terminal  of  the  dynamo.  This  reaction,  which  may  be 
written 

Ag-(-)=Ag+  (2) 

is  evidently  the  reverse  of  the  one  at  the  cathode  and  hence  the 
electrolysis  in  this  cell  consists  in  the  dissolving  of  metallic  silver 
from  the  anode  and  the  plating  out  of  metallic  silver  upon  the 
cathode. 


nvMAMO 


Fig.  28. 

Case  2. — Let  the  cell  in  Fig.  28  consist  of  a silver  anode  and  a 
silver  chloride  cathode  dipping  into  a solution  of  potassium 
chloride.  A silver  chloride  cathode  consists  simply  of  a silver 
wire  covered  with  a coating  of  solid  AgCl.  The  electrode 
processes  are  as  follows:  The  electrons  which  flow  to  the  cathode 


Sec.  4] 


THE  CONDUCTION  OF  ELECTRICITY 


195 


from  the  dynamo  terminal  are  removed  by  the  following  electro- 
chemical reaction: 

AgCl  + (-)  = Ag+Cl-  (3) 

that  is,  the  silver  chloride  is  reduced  to  metallic  silver,  the  chlo- 
rine atoms  taking  up  the  electrons  from  the  wire  and  passing  into 
solution  as  chloride  ions.  At  the  anode  the  reverse  reaction 
occurs, 

Cl-+Ag  = AgCl+(-)  (4) 

The  chloride  ions  give  up  their  electrons  to  the  wire  and  unite 
with  the  silver  electrode  forming  a coating  of  silver  chloride 
over  it. 

Case  3. — Let  the  cell  in  Fig.  28  consist  of  a platinum  anode 
and  a platinum  cathode  dipping  into  a solution  containing  a 
mixture  of  ferric  sulphate  (Fe2(SOi)3  <=*  2Fe+++  + 3S04  ) and 

ferrous  sulphate  (FeS04  <=±  Fe+++S04--).  The  electrons  com- 
ing to  the  cathode  from  the  dynamo  are  removed  from  the 
cathode  by  the  following  electrochemical  reaction: 

Fe++++(-)=Fe++  (5) 

that  is,  the  electrons  are  taken  up  by  the  ferric  ions  which  thereby 
lose  one  of  their  positive  charges  and  become  ferrous  ions.  At 
the  anode  the  reverse  reaction  occurs. 

(b)  Characteristic  Electrode  Potentials. — Let  the  cell  in  Fig. 
28  consist  of  a platinum  anode  and  a platinum  cathode  dipping 
into  a solution  containing  the  following  electrolytes:  KC1, 
Na2S04,  and  LiN03.  The  ions  arising  from  the  ionization  of 
these  salts  will  be  the  cations,  K+,  Na+  and  Li+  and  the  anions? 
Cl-,  S04  and  N03-.  In  addition  there  are  also  present  in 
every  aqueous  solution  very  small  quantities  of  hydrogen  ion 
H+  and  of  hydroxyl  ion  OH-  arising  from  the  ionization  of  the 
water  itself  which  takes  place  to  a very  slight  extent , according  to 
the  reaction 

H2O^H++OH-  (6) 

Now  when  the  electrons  coming  from  the  negative  terminal  of 
the  dynamo  begin  to  collect  upon  the  cathode  the  potential 
between  this  electrode  and  the  solution  thereby  increases  and 
the  negative  electrons  on  the  electrode  are  attracted  by  the 


196 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


positively  charged  ions  in  the  solution  in  contact  with  electrode. 
If  this  potential  difference  is  great  enough  and  is  continuously 
maintained  by  electrons  coming  from  the  dynamo,  we  have  seen 
that  a union  between  the  electrons  in  the  wire  and  one  or  more 
species  of  the  ions  in  the  solution  will  take  place.  In  other  words 
an  electrochemical  reaction  will  occur.  Now  in  the  present 
instance  there  are  obviously  at  least  four  electrochemical  re- 
actions, any  one  of  which  might  conceivably  take  place  at  our 
cathode,  namely: 


K+  + ( — ) = K (metal) 

(7) 

Na+-f-  ( — ) =Na  (metal) 

(8) 

Li++  ( — )=Li  (metal) 

(9) 

or 

2H++2(-)=H2  (gas) 

(10) 

Which  of  these  reactions  will  actually  occur? 

In  order  to  bring  about  a given  electrochemical  reaction  at  any 
electrode  we  must  first  create  a sufficient  potential  difference  in 
the  desired  direction,  that  is;  we  must  make  the  potential  dif- 
ference large  enough  to  force  the  electrons  to  go  in  the  direction 
desired.  Now  the  potential  difference  necessary  to  accomplish 
this  is  a characteristic  property  of  each  electrochemical  reaction 
and  of  the  concentrations  of  the  ions  and  molecules  involved  in 
that  reaction  as  well  as  of  the  thermodynamic  environment  by 
which  these  ions  and  molecules  are  surrounded.  This  potential 
difference  we  shall  call  the  characteristic  electrode  potential  of 
the  reaction.  More  explicitly,  the  characteristic  electrode  poten- 
tial of  a given  electrochemical  reaction  under  a given  set  of  con- 
ditions is  the  potential  difference  which  exists  betweep  the  elec- 
trode and  the  solution  when  the  two  are  in  equilibrium  with  each 
other  with  respect  to  the  reaction  in  question.  Whenever,  there- 
fore, the  potential  difference  at  any  electrode  is  kept  either  greater 
or  less  than  the  characteristic  potential  of  some  electrochemical 
reaction  which  is  possible  at  that  electrode,  then  this  reaction 
will  proceed  continuously  in  the  one  direction  or  the  other  as  long 
as  the  necessary  potential  difference  is  maintained. 

Now  in  the  present  instance,  as  the  potential  difference  be- 
tween our  cathode  and  the  solution  rises,  due  to  the  flow  of  elec- 
trons from  the  dynamo  to  the  cathode,  it  will  eventually  reach  a 
value  which  will  exceed  the  characteristic  electrode  potential  of 


Sec.  4] 


THE  CONDUCTION  OF  ELECTRICITY 


197 


some  one  of  the  four  possible  reactions  represented  by  equations 
(7)  to  (10).  When  this  happens  the  reaction  in  question  will  take 
place  at  our  cathode.  Of  these  reactions,  the  fourth  one  happens 
to  have  the  smallest  characteristic  electrode  potential  under  the 
given  conditions,  and  hence  the  electrolysis  of  our  solution  results 
in  the  evolution  of  hydrogen  gas  at  the  cathode  due  to  a de- 
composition of  the  water  as  represented  by  reactions  (6)  and  (10). 
These  may  be  added  together  and  written  thus, 

2H20+2(  — )=H2+20H~  (11) 

showing  that  hydroxyl  ion  is  produced  in  the  solution  as  one 
result  of  the  reaction.  In  other  words  the  solution  in  the 
neighborhood  of  the  cathode  becomes  alkaline.  If  a very  heavy 
current  is  passed  through  the  solution,  it  may  happen  eventually 
that  the  electrons  are  sent  to  the  cathode  from  the  dynamo 
faster  than  they  can  be  removed  by  reaction  number  (10).  In 
that  case  the  potential  at  the  electrode  will  rise  and  it  may  become 
great  enough  to  cause  reaction  (9)  to  take  place  to  some  extent 
simultaneously  with  reaction  (10),  since  its  characteristic  potential 
stands  next  above  that  of  reaction  (10) . If,  however,  any  metallic 
lithium  should  separate  on  the  electrode  as  a result  of  the  occur- 
rence of  reaction  (9),  it  would  immediately  proceed  to  react  with 
the  water,  thus 

2Li + 2H20  = 2Li+ + 20H~ + H2  (12) 

so  that  the  final  products  would  be  the  same  as  though  reaction 
(10)  were  the  only  one  which  occurred.  If  the  current  density 
(Le.,  amount  of  current  per  unit  electrode  surface)  at  the  elec- 
trode is  kept  small,  however,  only  reaction  (10)  will  take  place  at 
this  electrode. 

Let  us  now  turn  to  the  anode  of  our  cell  and  see  what  will 
happen  there.  The  reaction  must  of  course  be  one  which  will 
furnish  electrons  to  the  anode.  Any  one  of  the  following  would 
do  this: 

2C1~  = Cl2(gas)  +2(  — ) (13) 

Cr+3H20  = C108”+6H++6(-)  (14) 

S0r_  = S208"+2(-)  (15) 

40H-  - 02(gas)  +2H20 +4(  - ) (16) 


and 


198 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


Of  all  these  possible  reactions,  however,  the  last  one  has  the 
smallest  characteristic  electrode  potential  in  our  solution  and  is, 
therefore,  the  one  which  will  take  place  at  our  anode.  By  adding 
together  equations  (6)  and  (16)  we  obtain  the  equation 


2H20  = 4H++02+4(-) 


(17) 


which  represents  the  final  result  of  the  electrolysis  at  the  anode. 
Gaseous  oxygen  is  evolved  and  the  solution  around  the  anodo 
becomes  acid. 

The  electrode  products  of  the  electrolysis  with  this  cell  come, 
therefore,  entirely  from  the  water  itself  and  not  at  all  from  the 
dissolved  salts.  The  current  is  also  carried  from  electrode  to 
solution  and  vice  versa  entirely  by  the  ions  of  the  water  and  not 
to  any  appreciable  extent  by  the  ions  of  the  dissolved  electrolytes. 
The  current  through  the  solution,  however,  is  practically  all  carried 
by  the  ions  of  the  dissolved  electrolytes  because  there  are  so  few 
H+-  ions  and  OH_-ions  that  their  current  carrying  capacity 
(XVI,  3)  is  entirely  negligible.  The  reason  these  two  species 
of  ions  can  nevertheless  carry  all  of  the  current  between  the  elec- 
trode and  the  solution  is  because  at  the  electrodes,  as  fast  as 
these  ions  are  used  up  by  the  electrode  reaction,  a new  supply  is 
continuously  produced  by  the  ionization  of  some  more  water  ac- 
cording to  reaction  (6),  since  any  chemical  or  electrochemical 
reaction  can  be  made  to  proceed  continuously  in  one  direction 
provided  one  of  the  products  of  the  reaction  is  continually  re- 
moved or  destroyed  as  fast  as  it  is  formed,  and  this  is  what  hap- 
pens at  our  electrodes,  H+-ions  being  removed  at  the  cathode 
and  OH~-ions  at  the  anode. 

It  not  infrequently  happens  that  two  or  more  possible  elec- 
trochemical reactions  will,  in  a given  solution,  have  practically 
the  same  characteristic  electrode  potentials.  When  this  happens 
the  reactions  will  take  place  simultaneously  during  the  electrolysis 
and  a mixture  of  products  will  be  obtained.  This  happens,  for 
example,  during  the  electrolysis  of  a moderately  strong  solution 
of  HC1  with  platinum  electrodes.  At  the  anode  the  two  reactions 


2C1~  = Cl2(gas)  +2(  — ) 
and  40H-  = 02(gas)+2H20+4(-) 


(IS) 

(18a> 


;Sec.  4] 


THE  CONDUCTION  OF  ELECTRICITY 


199 


occur  simultaneously,  a mixture  of  oxygen  and  chlorine  gases 
being  evolved  at  the  electrodes.  Even  when  their  characteristic 
electrode  potentials  are  quite  different,  mixed  electrochemical 
reactions  can  be  made  to  occur  by  forcing  more  current  through 
the  cell  than  one  reaction  can  take  care  of,  as  in  the  example 
given  above  (p.  197).  The  actual  products  of  the  electrolysis 
of  any  given  solution  depend,  therefore,,  greatly  upon  the  condi- 
tions under  which  the  electrolysis  is  carried  out  and  a clean-cut 
single  electrochemical  reaction  entirely  free  from  even  traces  of 
other  reactions  is,  as  a matter  of  fact,  rather  difficult  to  bring 
about  in  any  prolonged  electrolysis.  In  Table  XIX  are  shown 
some  of  the  electrochemical  reactions  which  occur  during  electro- 
lyses of  aqueous  solutions  with  platinum  electrodes. 


Table  XlXa 


Principle  Electrode  Reactions  and  Products  of  the  Electrolysis  of  Aqueous 
Solutions  with  Platinum  Electrodes.  (M  = alkali  metal) 


Anode  reaction 

Solute 

Cathode  reaction 

AgN03 

CuS04 

Ag+  + ( — ) = Ag 
Cu+++2(-)=Cu 

2H20  = 4H++02+4(  — ) 

mno3 

m2so4 

; 2H20+2(-)=20H-+H2 

h2so4 

h3po4 

Dilute  HC1 
Dilute  HN03 

2H++2(  — )=H2 

2Cl-  = Cl2+2(  — ) 

Cone.  HC1 

40H-  = 2H20+02+4(-) 

MOH 

2H20+2(  — )=20H-+H2 

|2H20  =4H++02+4(  — ) 

Cone.  HN03 

NO,-+4H++3(-)  = 

3H20+  NO 

and 

N03-+9H++8(-)  = 
3H20+NH3 

2I-  = l2+2(  — ) 

mi+i2 

I‘2+2(  — ) =21“ 

200 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


Problem  1. — What  conditions  must  prevail  in  a solution  in  order  that  an 
object  which  is  made  the  cathode  in  the  solution  shall  be  plated  with  brass  by 
electrolysis  ? 

(c)  Voltaic  Action. — In  the  Daniell  cell  a copper  sulphate  solu- 
tion containing  a copper  electrode  is  placed  in  contact  (by 
means  of  a porous  wall  or  partition)  with  a zinc  sulphate  solution 
containing  a zinc  electrode.  At  each  electrode  there  exists  a 
potential  difference  between  solution  and  electrode  which  po- 
tential difference  is  the  characteristic  electrode  potential  at  the 
electrode  in  question  under  the  given  conditions,  the  two  electro- 
chemical reactions  involved  being  respectively 


Cu  — 2(  — ) = Cu++ 
and  Zn  — 2(  — )=Zn++ 

The  total  potential  across  the  terminals  of  this  cell  ( i.e .,  the 
potential  difference  between  the  two  electrodes)  will,  therefore, 
be  equal  to  the  algebraic  sum  of  the  two  characteristic  electrode 
potentials  plus  any  potential  difference  which  may  exist  at  the 
junction  between  the  two  solutions.  Now  in  this  particular 
cell  it  happens  that  the  two  characteristic  electrode  potentials 
are  such  that  the  total  potential  across  the  cell  is  about  1.1 
volts,  the  zinc  electrode  being  negative  with  respect  to  the  copper. 
If  the  two  electrodes  are  connected  externally  by  a wire,  a 
current  of  electricity  will,  therefore,  flow  through  the  wire  from 
the  copper  to  the  zinc,  and  zinc  will  dissolve  at  the  anode  and 
copper  deposit  on  the  cathode.  The  total  reaction  of  the  cell 
will  be  the  algebraic  difference  of  the  two  electrode  reactions, 
which  is 

Zn+Cu++  = Zn+++Cu  ' (19) 

When  two  electrodes  are  arranged  in  a cell  so  that  the  two  char- 
acteristic electrode  potentials  are  such  that  the  cell  is  capable, 
as  above,  of  producing  an  electric  current  in  an  external  circuit, 
the  current  is  said  to  be  produced  by  voltaic  action  and  the  cell 
is  a primary  battery.  Voltaic  action  and  electrolysis  by  means 
of  a current  produced  externally  and  forced  through  the  cell  are 
essentially  identical  phenomena,  however,  and  obey  the  same 
laws.  A reaction  such  as  that  represented  by  equation  (19) 


Sec.  4] 


THE  CONDUCTION  OF  ELECTRICITY 


201 


will  also  evidently  occur  without  the  production  of  an  external 
current  if  a piece  of  zinc  be  simply  dipped  into  a solution  of 
a cupric  salt  (Cf.  X,  9). 

(d)  Normal  Potentials  and  Normal  Electrodes. — For  a given 
cell,  each  of  the  two  characteristic  electrode  potentials  (which 
together  with  the  potential  at  the  liquid  junction  make  up  the 
total  E.M.F.  of  the  cell)  depends  upon  the  concentrations  of  the 
molecular  and  ion  species  which  take  part  in  the  electrochemical 
reaction  at  the  electrode,  as  well  as  upon  the  thermodynamic 
environment  in  the  solution  around  the  electrode.  In  writing 
the  equation  of  an  electrochemical  reaction  a line  above  a formula 
will  hereafter  be  employed  to  indicate  that  the  molecular  species 
^2  is  present  in  the  gaseous  state,  a line  below  the  formula  will  be 
]>  similarly  used  to  indicate  the  crystalline  state,  while  the  absence 
«^of  such  lines  will  indicate  that  the  molecular  or  ion  species  in 
Q question  is  present  only  in  solution.  The  so-called  “normal 
potential’ ’ of  a given  electrochemical  reaction  is  the  characteris- 
*Ftic  electrode  potential  for  that  reaction  when  all  gaseous  sub- 
5*stances  involved  in  the  reaction  as  written  are  present  under  a 
pressure  of  one  atmosphere  and  all  similarly  involved  molecular 
2and  ion  species  which  are  present  only  as  solutes  have  each  a 
-Lconcentration  of  one  formula  weight  per  liter.  Thus  the  poten- 
Qtial  of  the  “ normal  hydrogen  electrode”  is  the  potential  at  a 
’’^platinum  electrode  saturated  with  hydrogen  gas  under  a pressure 
of  one  atmosphere  dipping  into  a solution  containing  hydrogen 
ion  at  a concentration  of  one  formula  weight  per  liter  when 
equilibrium  is  attained  with  respect  to  the  reaction,  H2  — 2(  — ) = 
2H+;  the  potential  of  the  “ normal  zinc  electrode”  is  the  potential 
at  a zinc  electrode  dipping  into  a solution  containing  zinc  ion  at  a 
concentration  of  one  formula  weight  per  liter  when  equilibrium 
is  attained  with  respect  to  the  reaction  Zn  — 2(  — ) = Zn++;  the 
potential  of  a -“normal  calomel  electrode”  is  the  potential  of  a 
mercury  electrode  covered  with  a layer  of  crystalline  HgCl  in 
contact  with  a solution  containing  chloride  ion  at  a concentration 
of  one  formula  weight  per  liter  when  equilibrium  is  attained  with 
respect  to  the  reaction,  Hg-f  Cl~—  ( — ) =HgCl;  and  the  poten- 
tial of  a “normal  arsenious-arsenic  electrode  is  the  potential  at 
any  suitable  electrode  which  is  in  electrochemical  equilibrium 


202 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


with  respect  to  the  reaction,  H3As03  + H20  — 2(  — ) =H3As04+ 
2H+,  when  each  of  the  solute  species  is  present  at  a concentration 
of  one  formula  weight  per  liter;  etc. 

By  joining  together  the  solutions  surrounding  any  two  normal 
electrodes  by  means  of  some  suitable  intermediate  liquid  (such  as 
a solution  of  KN03)  a voltaic  cell  is  formed  (Cf.  above,  4c)  and 
the  total  electromotive  force  of  such  a cell  will  (except  for  the 
small  potentials  at  the  liquid  junctions  between  the  two  electrode 
vessels)  be  equal  to  the  algebraic  sum  of  the  normal  potentials  of 
the  two  electrodes.  If  one  of  the  electrodes  of  such  a cell  is  the 
normal  hydrogen  electrode,  then  the  total  electromotive  force  of 
the  cell  (after  subtracting  the  liquid  junction  potentials)  is 
arbitrarily  taken  as  the  normal  potential  of  the  other  electrode, 
that  is,  in  computing  the  values  of  the  different  normal  electrode 
potentials  the  potential  of  the  normal  hydrogen  electrode  is 
arbitrarily  taken  as  zero. 

In  Table  XIX6  are  given  the  “ hypothetical  normal’’  potentials 
of  a number  of  electrochemical  reactions  based  upon  the  normal 
hydrogen  electrode  as  zero.  They  are  termed  “hypothetical 
normal”  because  they  have  been  computed  from  measurements 
at  various  concentrations  on  the  assumptions:  (1)  that  the 
equivalent  conductance  ratio  (see  XVII,  2b,  equation  10)  is  a 
correct  measure  of  the  degree  of  ionization  of  all  electrolytes 
even  for  concentrations  greater  than  normal;  and  (2)  that  the 
thermodynamic  environment  in  all  solutions  even  for  concen- 
trations greater  than  one  mole  per  liter  is  the  same  as  it  is  in 
pure  water;  in  other  words  the  laws  of  dilute  solutions  (XIII,  3) 
are  assumed  to  hold  for  all  such  solutions.  Neither  of  these  two 
assumptions  can  be  regarded  as  correct,  hence  the  expression 
“hypothetical  normal”  as  applied  such  values  as  those  given  in 
Table  XIX6.  We  are  not  at  present  in  a position  to  calculate 
the  true  normal  potentials  of  electrochemical  reactions  and 
in  view  of  this  fact,  it  would  be  better  to  adopt  some  other  basis 
(milli-normal  potentials,  for  example)  for  computing  and  record- 
ing characteristic  electrode  potentials,  but  this  has  not  as  yet 
been  done.  A more  complete  table  of  “hypothetical  normal” 
potentials  can  be  found  in  a monograph  by  Abegg,  Auerbach  and 
Luther. 


Sec.  4] 


THE  CONDUCTION  OF  ELECTRICITY 


203 


Table  XIX6 

So-called  “Normal  Potentials”  for  some  typical  electrochemical  reactions, 
based  upon  the  “normal”  hydrogen  electrode  as  unity.  The  reaction  tends 
to  go  in  the  direction  indicated  and  charges  the  electrode  with  a charge  of 
the  sign  and  magnitude  given. 


Electrochemical 

reaction 

volts 

Electrochemical  reaction 

+ 

volts 

Li  — ( — ) =Li+ 

2.74 

2H++2(-)=h: 

±0.0 

K-(-)=K+ 

2.64 

HgCl  + (-)=Hg(liq)+Cl- 

0.275 

Na  — ( — )=Na+ 

2.43 

Cu+++2(-)=Cu 

0.34 

Mg— 2(  — ) = Mg++ 

1.55 

Ag(NH3)2++(-)  = Ag+2NHs 

0.38 

Zn  — 2(  — ) =Zn++ 

0.76 

+2H20 +4  ( - ) = 40H- 

0.41 

Fe-2(-)=Fe++ 

o 

CO 

H-f2(— )=2I~ 

j 0.54 

Cd— 2(-)=Cd++ 

0.40 

Fe+++  + ( — ) =Fe++ 

0.75 

Co— 2(  — ) =Co++ 

0.3 

Ag+  + (-)=Ag 

0.80 

Ni-2(-)=Ni++ 

0.2 

Hg+  + (-)=Hg(liq) 

0.80 

Pb  — 2(  — ) =Pb++ 

0.12 

Br2+2(  — ) =2Br~ 

1.10 

Sn  — 2(  — ) =Sn++ 

0.10 

ClI+2(-)=2Cl- 

1.35 

Fe  — 3 ( — ) = Fe+++ 

0.04 

Au+  + ( — ) = Au 

1.5 

H;-2(-)=2H+ 

+ 0.0 

Mn04-+8H++5(  — ) =Mn++  + 
4H20 

1.52 

Problem  2. — How  could  you  arrange  the  cell  so  as  to  obtain  an  electric 
current  from  the  occurrence  of  each  of  the  following  reactions  in  the 
direction  indicated: 

Fe+Cu++  = Fe+++Cu 
Zn+2H+  = Zn+++H2 
3Zn+8H++2N03-  = 3Zn+++2N0+4H20 
3Zn+Cr207--+7H+  = 3Zn+++2Cr++++7H20 

From  the  data  in  Table  XIX5  calculate  approximately  the  potential  of 
the  cell  in  the  first  three  cases.  Predict  also  from  the  data  in  this  table  what 


204 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


will  occur:  (a)  when  a piece  of  copper  is  dipped  into  a normal  AgN03  solu- 
tion; (b)  when  a piece  of  tin  is  dipped  into  a normal  FeCl3  solution;  (c)  when 
a piece  of  tin  is  dipped  into  a normal  HC1  solution;  (d)  when  a piece  of  copper 
is  dipped  into  a normal  HC1  solution;  (e)  when  a piece  of  silver  is  dipped  into 
a normal  Pb(N03)2  solution;  (f)  when  a 2 N solution  of  FeCl2  is  mixed  with  a 
solution  of  HMn04,*  (g)  when  solid  iodine  is  shaken  with  a normal  solution  of 
KBr;  (h)  when  pure  gaseous  chlorine  is  passed  through  a normal  solution 
of  KI. 

Every  chemical  reaction  occurring  in  a solution  can  always  be 
regarded  as  made  up  of  two  opposing  electrochemical  reactions 
and  it  can  be  shown  to  follow  from  the  laws  of  thermodynamics 
that  whenever  equilibrium  (I,  9 and  XXII,  1)  has  been  attained 
in  such  a reaction,  the  concentrations  of  all  of  the  solute  species 
involved  must  be  such  that  the  characteristic  electrode  potentials 
of  the  two  opposing  electrochemical  reactions  are  equal  to  each 
other. 

The  brief  treatment  of  the  subject  of  characteristic  electrode 
potentials  in  this  section  and  the  problems  illustrating  the  sig- 
nificance and  use  of  electromotive  force  data  are  given  at  this 
point  in  order  that  the  student  may  appreciate  the  general  nature 
of  the  processes  involved  in  the  passage  of  electricity  across  a 
boundary  between  a metallic  and  an  electrolytic  conductor.  In 
practice  it  frequently  happens  that  owing  to  the  slowness  of 
some  electrochemical  reactions  the  actual  reactions  which  occur 
in  electrochemical  processes  are  not  always  those  which  one 
would  predict  from  such  data  as  those  given  in  Table  XIX6. 
The  phenomena  of  “ passivity”  and  “ over-voltage’’  and  many 
other  factors  and  relationships  which  must  be  understood  in  order 
to  handle  intelligently  electrochemical  problems  are  treated  at 
length  in  books  devoted  to  the  special  field  of  electrochemistry, 
a field  which  will  not  be  included  within  the  scope  of  this  book. 

5.  Faraday’s  Law  of  Electrolysis. — From  the  preceding  dis- 
cussion of  the  mechanism  of  electrolysis  it  is  clear  that  the 
chemical  changes  which  take  place  at  the  electrodes  are  the  re- 
sult of  the  taking  on  or  giving  up  of  electrons  by  atoms,  atom 
groups  and  ions  and  that  for  every  ion  involved  in  this  electrode 
process,  one,  two,  or  three,  etc.,  electrons  are  also  involved 
according  as  the  ion  carries  one,  two  or  three,  etc.,  charges  which 
must  be  neutralized,  that  is,  according  as  the  change  in  valence 


Sec.  5] 


THE  CONDUCTION  OF  ELECTRICITY 


205 


is  one,  two  or  three,  etc.,  units.  Hence  for  an  electrochemical 
change  involving  one  equivalent  weight  of  a substance  the  same 
quantity  of  electricity  will  always  be  required  and  this  quan- 
tity will  be 

g = Ne  (20) 

or  for  Ne  equivalents  of  chemical  change  the  quantity  of  electric- 
ity required  will  be 

q = KeNe  (21) 

where  N is  Avogadro’s  number  and  e,  the  charge  carried  by  one 
electron,  is  called  the  elementary  charge  of  electricity. 

If  we  put 

Ne  = F (22) 

equation  (21)  becomes 

q = FNe  (23) 

where  F is  a universal  constant.  The  significance  of  this  equa- 
tion may  be  stated  in  words  as  follows:  Whenever  an  electric 
current  passes  across  a junction  between  a purely  metallic  and 
a purely  electrolytic  conductor  a chemical  change  or  chemical 
changes  occur  the  amount  of  which,  expressed  in  chemical  equiva- 
lents, is  exactly  proportional  to  the  quantity  of  electricity  which 
passes  and  is  independent  of  everything  else.  This  statement 
is  known  as  Faraday’s  Law  of  Electrolysis  and  was  discovered 
by  Michael  Faraday  in  1833.  The  proportionality  constant, 
F,  in  equation  (23)  is  known  as  one  faraday  or  one  equivalent 
of  electricity  and  is  evidently  the  amount  of  electricity  re- 
quired to  produce  one  equivalent  of  chemical  change.  The 
quantities,  e and  F (equation  22),  evidently  bear  the  same  rela- 
tion to  each  other  that  the  weight  of  an  atom  does  to  the  atomic 
weight  of  the  element  (I,  7). 

The  value  of  the  faraday  has  been  very  accurately  determined 
by  measuring  (with  a silver  coulometer)  the  quantity  of  electric- 
ity required  to  deposit  one  equivalent  weight  of  metallic  silver 
from  a silver  nitrate  solution  and  also  by  measuring  (with  an 
iodine  coulometer)  the  amount  required  to  liberate  one  equiva- 
lent weight  of  iodine  from  a solution  of  potassium  iodide.  The 
result  obtained  was  F = 96,500  (±  0.01  per  cent.)  coulombs 
per  equivalent. 


206 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


An  apparatus  arranged  for  the  measurement  of  the  amount  of 
a given  chemical  change  produced  by  an  electric  current  is 
termed  a coulometer  (or  sometimes  a voltameter)  and  is  em- 
ployed for  measuring  the  quantity  of  electricity  passed  through 
a circuit.  The  two  most  accurate  types  are  the  silver  coulonieter 
and  the  iodine  coulometer.  In  the  former  metallic  silver  is 
deposited  electrolytically  from  a solution  of  silver  nitrate  and 
the  weight  of  the  deposit  determined.  In  the  latter  iodine  is 
liberated  electrolytically  from  a solution  of  potassium  iodide, 
the  amount  liberated  being  determined  by  titration. 

6.  The  Elementary  Charge  of  Electricity. — If  either  e or  N 
can  be  directly  measured  it  is  clear  that  the  other  one  can  then 
be  calculated  by  means  of  equation  (22),  since  F is  known. 

When  X-rays  pass  through  air  or  any  other  gas,  the  gas  be- 
comes ionized  (I,  2g),  that  is,  some  of  its  molecules  lose  tempo- 
rarily one  or  more  of  their  electrons  and  thus  acquire  a positive 
charge.  These  charged  molecules  are  called  gaseous  ions.  These 
ions  as  well  as  the  electrons  liberated  from  them  all  possess  the 
unordered  heat  motion  (II,  1)  of  the  other  molecules  of  the  gas, 
that  is,  they  dart  rapidly  about  in  all  directions  colliding  with 
anything  which  may  come  in  their  path.  If  a tiny  droplet  of  oil 
or  mercury  or  some  other  liquid  is  introduced  into  an  ionized 
gas,  it  will  eventually  collide  with  one  of  these  rapidly  moving 
ions  and  will  capture  and  hold  it.  The  droplet  thus  acquires  a 
charge  of  electricity  exactly  equal  to  that  possessed  by  the 
captured  ion  and  from  the  observed  behavior  of  the  charged  drop- 
let in  an  electric  field,  the  magnitude  of  its  charge  can  be  measured. 
In  this  way  Millikan  has  succeeded  in  obtaining  a very  accurate 
measurement  of  the  value  of  the  elementary  charge  of  electricity. 
His  apparatus  is  shown  in  Fig.  29. 

By  means  of  an  atomizer  A a fine  spray  of  oil  or  other  liquid 
is  blown  into  the  chamber  D and  one  of  the  droplets  eventually 
falls  through  the  pin  hole  p into  the  space  between  the  two  con- 
denser plates,  M and  N.  The  pin  hole  is  then  closed  and  the 
air  between  the  plates  is  ionized  by  a beam  of  X-rays  from  X. 
The  droplet  is  illuminated  through  the  windows  g and  c by  a 
powerful  beam  of  light  from  an  arc  at  a,  and  is  observed  through 
a telescope  directed  at  a third  window  not  shown  in  the  figure. 
The  droplet  eventually  acquires  a charge  by  collision  with  one 


Sec.  6] 


THE  CONDUCTION  OF  ELECTRICITY 


207 


of  the  ions  of  the  gas  and  can  then  be  caused  to  move  upward 
or  downward  at  will  by  charging  the  plates  M and  N with  electric 
charges  of  the  desired  signs,  by  means  of  the  switch  S.  The  speed 
of  the  drop  under  the  influence  of  the  electric  field  is  directly 
proportional  to  the  magnitude  of  its  charge,  and  the  observer 
watching  the  drop  observes  sudden  changes  in  its  speed  due  to  a 
change  in  the  charge  of  the  drop  brought  about  by  the  capture 


Fig.  29. — Millikan’s  apparatus  for  determining  the  value  of  e. 


now  and  then  of  an  additional  ion  with  which  it  happens  to 
collide.  By  measuring  this  change  in  the  speed  of  the  drop  im- 
mediately after  the  capture  of  an  ion  and  combining  it  with  the 
speed  of  the  drop  when  falling  under  the  influence  of  gravity, 
the  magnitude  of  the  charge  of  the  captured  ion  can  be  calcu- 
lated. From  a very  large  number  of  experiments  of  this  kind 
Professor  Millikan  found  that  the  charge  carried  by  a gaseous 
ion  is  always  equal  to  or  is  a small  exact  multiple  of  1.59  X10~19- 
( + 0.2  per  cent.)  coulombs  and  this  is,  therefore,  the  elementary 
charge  of  electricity,  the  charge  of  one  electron.  Substituting 
this  value  for  e in  equation  (22)  above  we  find  for  N the  value 


208 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVI 


N = 6.062T023(  + 0.2  per  cent.),  and  this  method  is  the  most  ac- 
curate one  which  we  have  at  present  for  determining  the  value 
of  Avogadro’s  number.  (Cf.  IX,  4.) 

7.  Electrical  Resistance  and  Conductance.  Ohm’s  Law. — In  a 
conductor  through  which  a steady  direct  current  is  passing,  the 
strength  of  the  current  or  simply  the  current,  7,  is  defined  by  the 
equation 

I=j  (24) 

where  q is  the  quantity  of  electricity  which  passes  through  a given 
cross  section  of  the  conductor  in  the  time  t.  By  direct  current  is 
meant  one  in  which  the  “ direction  of  the  current”  (XVI,  2)  is 
constant.  A current  whose  direction  periodically  varies  is  called 
an  alternating  current. 

According  to  Ohm’s  Law  and  the  nature  of  our  definitions  of 
electrical  quantities,  the  value  of  a direct  current  through  any 
homogeneous  metallic  or  electrolytic  conductor  is  equal  to  the 
electromotive  force  (E.M.F.),  E,  between  the  ends  of  the  con- 
ductor, divided  by  the  resistance,  R,  of  the  conductor,  or 

7=^  (Ohm’s  Law)  (25) 

The  reciprocal  of  the  resistance  is  called  the  conductance,  L. 

lJr  (26> 


The  unit  of  conductance  is  called  the  reciprocal  ohm  or  the  mho. 

The  conductance,  L,  of  any  homogeneous  conductor  of  uni- 
form cross  section  is  proportional  to  the  area,  A,  of  the  cross 
section  and  inversely  proportional  to  the  length,  l,  of  the  con- 
ductor, or 


The  proportionality  constant,  L,  is  evidently  the  conductance  of 
a conductor  1 cm.  long  and  1 sq.  cm.  in  cross  section.  It  is 
called  the  specific  conductance  of  the  material  of  which  the  con- 
ductor is  composed.  It  depends  upon  the  chemical  nature  of 
the  material,  its  temperature,  pressure  and  physical  condition. 


Sec.  7] 


THE  CONDUCTION  OF  ELECTRICITY 


209 


The  reciprocal  of  the  specific  conductance  is  called  the  specific 
resistance,  R. 

R = Rj  (28) 

In  dealing  with  the  conductance  of  solutions  of  electrolytes  an 
additional  quantity  known  as  the  equivalent  conductance  is 
also  employed.  The  equivalent  conductance,  A,  of  any  solu- 
tion of  an  electrolyte  is  defined  as  the  conductance  of  that  volume 


of  the  solution  which  contains  one  equivalent  weight  of  the  elec- 
trolyte when  measured  between  two  parallel  electrodes  1 cm. 
apart.  (See  Fig.  30.) 

Problem  1. — Show  that  the  specific  conductance  and  the  equivalent  con- 
ductance of  a solution  containing  C equivalents  of  electrolyte  per  liter  are 
by  the  nature  of  their  definitions,  connected  by  the  relation, 


1000L 

A=  c 


(29) 


If  the  concentration  C is  expressed  in  moles  per  liter,  the  quantity  defined 
by  equation  (29)  is  called  the  molal  or  the  molecular  conductance,  and  is 
then  usually  represented  by  the  symbol,  X,  in  case  the  molal  and  equivalent 
weights  are  different. 

Problem  2. — A cylindrical  resistance  cell  2 cm.  in  diameter  is  fitted  with 
horizontal  silver  electrodes  4 cm.  apart,  and  is  filled  with  0.1  normal  silver 
nitrate  solution.  An  E.M.F.  of  0.5  volt  causes  3.72  milliamperes  to  flow 
through  the  solution.  Calculate  the  conductance,  the  specific  conductance, 
and  the  equivalent  conductance. 


210 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVr 


Problem  3. — How  much  copper  will  be  deposited  electrolytically  in  1 
hour  by  a current  which  produces  0.1  gram  of  hydrogen  in  the  same  time? 
What  is  the  average  strength  of  the  current  in  amperes? 

Problem  4. — How  much  copper  would  be  precipitated  from  a cuprous 
chloride  solution  in  1 hour  by  a current  which  produces  0.1  gram  of  hydro- 
gen in  the  same  time? 

Problem  5. — How  many  coulombs  of  electricity  will  be  required  in  order 
to  deposit  10  grams  of  silver  in  60  minutes  from  a solution  of  silver  nitrate? 
How  many  will  be  required  in  order  to  deposit  the  same  amout  of  silver  in 
135  minutes? 

Problem  6. — If  a current  of  0.2  ampere  is  passed  for  50  minutes  through 
dilute  sulphuric  acid  between  platinum  electrodes,  what  volume  of  hydrogen 
and  what  volume  of  oxygen,  measured  dry  at  27°  and  1 atmosphere,  will  be 
produced? 

Problem  7. — A silver  electrode  weighing  x grams  is  made  the  anode  in  a 
hydrochloric  acid  solution  during  the  passage  of  a current  of  0.1  ampere  for 
10  hours.  After  drying,  the  electrode  together  with  the  closely  adherent 
layer  of  silver  chloride  is  found  to  weigh  10  grams.  Calculate  x. 

Problem  8. — A cupric  sulphate  solution  is  electrolyzed  with  a copper 
cathode  and  platinum  anode  until  6.35  grams  of  copper  are  deposited.  The 
gas  evolved  at  the  anode  is  measured  at  25°  over  water  whose  vapor  pressure 
at  this  temperature  is  23.8  mm.  Barometric  pressure  =752  mm.  What 
volume  does  it  occupy?  What  is  the  mass  of  the  gas  evolved?  How  much 
time  would  be  required  for  the  electrolysis  if  a current  of  0.1  ampere  were 
used? 

Problem  9. — A current  of  electricity  is  passed  for  53  hours  through  an 
aqueous  solution  of  Na2S04  between  platinum  electrodes.  A silver  coul- 
ometer  connected  in  series  shows  a deposit  of  10.8  grams  of  silver  in  this 
time.  The  gas  evolved  at  the  cathode  in  the  Na2S04  solution  is  collected 
and  measured  dry  at  20°  under  a pressure  of  756  mm.  (1)  What  volume  will 
it  occupy?  (2)  What  is  its  mass?  (3)  What  was  the  average  current  strength 
employed  in  the  electrolysis? 

Problem  10. — A solution  of  cupric  sulphate  containing  5 grams  of  copper 
is  allowed  to  stand  in  contact  with  an  excess  of  bright  iron  tacks,  until  all  of 
the  copper  has  been  deposited.  If  the  solution  is  now  boiled  with  bromine 
water  in  excess  and  then  with  an  excess  of  NH4OH  and  the  resulting  pre- 
cipitate filtered  off,  ignited,  and  weighed,  how  much  will  it  weigh? 

Problem  11.  —Twenty  grams  of  bright  copper  gauze  are  allowed  to  stand 
in  contact  with  1 liter  of  a 0.05  normal  silver  sulphate  solution  until  action 
ceases.  The  gauze  is  then  removed,  dried  and  weighed.  How  much  does  it 
weigh?  The  solution  remaining,  which  may  be  assumed  to  contain  only 
cupric  sulphate,  is  electrolyzed  between  platinum  electrodes  in  such  a 
manner  that  copper  only  is  liberated  at  the  cathode.  How  much  copper  is 
deposited?  The  gas  given  off  at  the  anode  is  mixed  with  an  excess  of  hydro- 
gen (x  grams)  and  exploded.  The  residual  gas  is  measured  over  water  at 
20  and  772  mm.  and  found  to  occupy  a volume  of  2240  c.c.  Calculate  x. 
The  vapor  pressure  of  water  at  20°  is  17.5  mm. 


CHAPTER  XVII 


CONDUCTANCE  AND  DEGREE  OF  IONIZATION 

1.  Equivalent  Conductance  and  Concentration,  (a)  Ao  Values. 

— Let  one  equivalent  weight  of  an  electrolyte  dissolved  in  water 
be  placed  between  two  parallel  electrodes  1 cm.  apart  and  of 
indefinite  area,  as  shown  in  Fig.  30.  The  conductance  of  this 
solution  measured  in  this  cell  is  by  definition  (XVI,  7)  its 
equivalent  conductance.  If  the  solution  in  the  cell  be  now 
gradually  diluted,  the  conductance  will  be  observed  to  change 
and  will  eventually  increase  gradually  and  continuously  and 
approach  a definite  limiting  value  at  infinite  dilution.  This 
behavior  is  illustrated  by  the  data  in  Table  XX. 

Table  XX 


Illustrating  the  variation  of  equivalent  conductance  with  concentration  in 
aqueous  solution  at  18° 


C,  (equiv.  per  liter)  = 

1.0 

0.5 

0.1 

0.03 

0.01 

0.001 

0.0001 

0.00 

KC1 = 

98.27 

102.41 

112.03 

.... 

122.43 

127.34 

129.07 

129.5 

AgN03 = 

67.6 

77.5 

94.33 

107.80 

113.14 

115.01 

115.7 

NH4OH = 

0.89 

1.35 

3.10 

5.66 

9.68 

28 

? 

(238) 

.hc2h3o2 = 

1.32 

2.01 

4.68 

8 . 52 

14.59 

41 

107 

(348) 

In  the  case  of  the  first  two  electrolytes  (strong  electrolytes, 
XV,  4)  in  this  table  the  limiting  value,  Ao,  of  the  equivalent  con- 
ductance shown  in  the  last  column  is  obtained  by  extrapolation 
from  the  preceding  values,  and  although  extrapolation  is  always 
a more  or  less  uncertain  process,  the  values  obtained  for  strong 
electrolytes  are  usually  fairly  reliable  (to  1 per  cent,  or  better  in 
most  cases)  because  the  extrapolation  extends  over  a short  dis- 
tance only,  that  is,  the  extrapolated  value  is  not  greatly  different 
from  the  last  measured  value.  In  the  case  of  the  last  two  electro- 
lytes (weak  electrolytes,  XV,  4),  however,  the  Ao  values  cannot 

211 


212 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVII 


be  obtained  in  this  manner  owing  to  the  very  long  extrapola- 
tion which  would  be  necessary,  for  conductance  measurements  at 
concentrations  below  0.0001  normal  are  exceedingly  difficult 
to  carry  out  with  any  accuracy.  The  Ao  values  for  weak  electro- 
lytes are,  therefore,  obtained  by  another  method  described  below. 

(b)  Ion  Conductances. — The  increase  of  A with  dilution  is 
interpreted  by  the  Ionic  Theory  as  follows:  With  increasing 
dilution  the  degree  of  ionization(XV,  4)  of  the  electrolyte  increases 
and  since  this  results  in  a larger  number  of  ions  (current  carriers) 
between  the  electrodes  of  the  cell  the  conductance  increases 
correspondingly.  When  the  electrolyte  is  completely  ionized 
(as  at  infinite  dilution)  its  equivalent  conductance  in  the  solu- 
tion, A0,  is  evidently  the  equivalent  conductance  of  a solution 
containing  one  equivalent  weight  of  each  of  the  ions.  The 
magnitude  of  A0  for  an  electrolyte  will  obviously  depend  upon 
the  speeds  with  which  the  ions  of  that  electrolyte  move  through 
the  water,  since  the  faster  they  move  the  more  efficient  they  are 
as  carriers  of  electricity,  and  for  a given  electrolyte  A0  will 
obviously  be  made  up  of  the  separate  equivalent  conductances 
of  the  ions  of  that  electrolyte.  Thus  the  A0  values  of  the  follow- 
ing electrolytes  at  18°  are  expressed  by  the  equations 

NaCl,  A0=AoNa++Aocl_  =43.2+65.3  = 108.5 
LiOH,  Ao  = AoLi++AoOH_  =33.2+174  =207 
HNO3,  A0  = Aoh++AoNq3__  =313  + 61.7  =375 

the  conductances  being  expressed  in  reciprocal  ohms. 

Table  XXI  shows  the  equivalent  conductances  at  infinite 
dilution  of  some  of  the  more  important  ions  at  18°  together  with 
their  temperature  coefficients.  The  method  by  whicih  the  in- 
dividual ion  conductances  are  obtained  from  the  A0  values  of  the 
electrolytes  will  be  discussed  in  the  next  chapter.  By  adding 
together  the  proper  ion  conductances  one  can  obtain  the  A0 
value  for  any  electrolyte.  This  statement  is  known  as  Kohl - 
rausch’s  law  of  the  independent  migration  of  ions.  This  law  is 
especially  important  in  the  case  of  weak  electrolytes  where  the 
A0  values  cannot  be  obtained  by  extrapolation  from  the  con- 
ductance data  because  even  at  high  dilution  (0.0001  normal) 
the  ionization  is  still  far  from  complete.  (Cf.  above  and  Table 
XX.) 


Sec.  1]  CONDUCTANCE  AND  DEGREE  OF  IONIZATION 


213 


Table  XXI 


Ion  conductances  at  infinite  dilution  and  their  temperature  coefficients,  at 
18°.  Based  upon  the  measurements  of  Kohlrausch.  (See  Bates,  Jour. 
Amer.  Chem.  Soc.,  35,  534  (1913).) 


Cations 

Anions 

Ion 

A°ch 

A (di),8° 

Ion 

Ao„- 

1 (dAi 

A \dt ) i8° 

H+ 

313. 

! 0.0154 

OH- 

174. 

0.018 

Cs+ 

67.4 

0.0212 

ci- 

65.3 

0.0216 

K+ 

64.3 

0.0217 

Br- 

67.2 

0.0215 

nh4+ 

64.5 

0 . 0222 

I- 

65.9 

0.0213 

Na+ 

43.2 

0.0244 

N03- 

61.7 

0.0205 

Li+ 

33.2 

0.0265 

CIO3- 

54.8 

0.0215 

T1+ 

65.5 

0.0215 

Br03- 

48. 

Ag+ 

54.0 

0 . 0229 

KV 

33.9 

0.0234 

£Ca++ 

51.0 

0.0247 

c2h3o2- 

35. 

0.0238 

45.0 

0.0256 

KhCq 

61. 

0.0231 

|Ba++ 

55.0 

0.0239 

ISOr- 

67.7 

0.0227 

£Pb++ 

61.0 

0.0240 

^Cr04__ 

72. 

Problem  2. — Compute  the  A0  values  for  the  following  electrolytes  at 
18°  from  the  individual  ion  conductances:  oxalic  acid,  ammonium  bromide, 
barium  iodate,  ammonium  hydroxide,  acetic  acid.  The  A0  value  for 
§ZnS04at  18°  is  113.0  reciprocal  ohms.  Calculate  A0iZn++. 


(c)  Empirical  Dilution  Laws. — For  expressing  the  rate  of 
change  of  the  equivalent  conductance  of  an  electrolyte  with  its 
concentration  a number  of  empirical  equations  have  been  pro- 
posed and  used  by  various  investigators.  Of  these  relations  we 
shall  mention  here  only  two  of  the  most  recent  ones.  An  equa- 
tion proposed  by  Kraus  has  been  tested  by  Kraus  and  Bray  for  a 
great  variety  of  electrolytes  in  many  different  kinds  of  solvents 
and  has  been  found  to  be  on  the  whole  very  satisfactory.  It  has 
the  following  form: 


2 C 


Ay  \ 
A^yJ 


— k -(-  k 


,/CAyy 

\Aq7]qJ 


(la) 


where  77/ 770  is  the  ratio  of  the  viscosity  (III,  6)  of  the  solution  to 
that  of  water  at  the  same  temperature  and  A^/rjo  is  called  the 
“corrected”  equivalent  conductance  (Cf.  Sec.  2b  below).  The 
quantities  k,  k',  h and  likewise  A0  are  empirical  constants  whose 


214 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVII 


values  are  so  chosen  as  give  the  best  agreement  between  the  cal- 
culated and  observed  conductance  values. 

In  the  last  column  of  Table  XXII  are  shown  the  values  of 
A V/t) o calculated  from  the  Kraus  equation  for  KC1  solutions  at 
18°,  using  for  the  four  empirical  constants  the  values  given  at 
the  head  of  the  table.  On  comparing  these  values  with  the  ob- 
served values  for  the  same  concentrations  as  shown  in  column  4, 
it  will  be  seen  that  the  agreement  is  very  good  except  for  concen- 
trations below  0.001  molal.  Here  the  equation  seems  to  break 
down  and  apparently  leads  to  a A0  value  much  lower  than  the 
ordinarily  accepted  one  for  this  salt. 

Bates  has  proposed  an  equation  of  the  same  form  as  that  of 
Kraus  except  that  the  logarithm  of  the  left-hand  expression  is 
employed  instead  of  the  expression  itself. 


The  constants  k,  k'  and  h in  the  Bates  equation  are  empirical 
ones  to  be  evaluated  from  the  experimental  data,  but  the  constant 
A0  represents  the  value  of  A at  infinite  dilution,  obtained  by 
extrapolation  from  the  most  reliable  measurements  at  high  dilu- 
tions without  reference  to  the  values  of  the  other  constants  in 
the  general  equation  or  to  the  form  of  this  equation.  In  column 
5 of  Table  XXII  are  shown  the  values  of  A calculated  from 
the  Bates  equation  for  KC1  solutions  at  18°  using  for  the  three 
empirical  constants  k,  k'  and  h,  the  values  given  at  the  head  of 
the  table.  The  value  for  A0  is  that  obtained  from  Table  XXI. 
The  agreement  between  these  calculated  values  and  the  observed 
values  is  excellent  throughout  the  whole  concentration  range. 

For  moderate  concentrations  the  viscosity  ratio  ^/rj  0 may  be 
taken  as  unity  and  for  weak  electrolytes  at  small  concentrations 

the  product  — ^ is  so  small  that  the  term  containing  it  on  the 
Ao 

right  hand  side  of  the  equation  is  usually  negligible  in  comparison 
with  the  term  k and  hence  for  weak  electrolytes  both  equations 
reduce  to  the  form 


A 2C 

Ao  (Ao  - A)  = 


(lc) 


Sec.  1]  CONDUCTANCE  AND  DEGREE  OF  IONIZATION 


215 


which  is  known  as  Ostwald’s  dilution  law  for  weak  electrolytes. 
It  expresses  the  results  for  such  electrolytes  with  great  exactness 
and  we  shall  see  later  (XXII,  Table  XXVII)  that  it  can  be 
deduced  theoretically  from  the  Second  Law  of  Thermodyna- 
mics and  the  Solution  Laws. 


> 


Table  XXII 


A comparison  of  the  empirical  equations  of  Bates  and  of  Kraus  with 
Kohlrausch’s  conductance  data  for  KC1  at  18°. 


Bates’  equation:  Logio 


where  Ao,  k,  k',  and  h are  empirical  constants  whose  values  for  KC1  at  18° 
are  129.50,  —3.5905,  4.020,  and  0.0775  respectively.  (Bates,  Jour.  Amer. 
Chem.  Soc.  37,  0000  (1915)). 


Kraus’  equation: 


k + k' 


CAv\h 

AoVoJ 


where  Ao,  k,  k'}  and  h are  empirical  constants  whose  values  for  KC1  at  18° 
are  128.3,  0.080,  2.707,  and  0.763  respectively.  (Kraus  and  Bray,  Jour. 
Amer.  Chem.  Soc.  35,  1412  (1913)). 


Concentra-  j 
tion,  C moles 
per  liter 

Relative 
viscosity 
of  the 
solution, 
rj/vo 

Equivalent  con- 
ductance observed 

Equivalent 

conductance 

calculated 

(Bates) 

Av/vo 

Equivalent 
conduct- 
ance  cal- 
l culated 
(Kraus) 

A v/vo 

A 

A v/vo 

3. 

0.9954  j 

88.3 

87.89 

87.9 

87.4 

2. 

0.9805  ! 

92.53 

90.73 

91.1 

90.9 

1. 

0.982 

98.22 

96.5 

96.53 

96.4 

0.5 

0.9898 

102.36 

101.32 

101.29 

101.1 

0.2 

0.9959 

107.90 

107.46 

107.43 

107.6 

0.1 

0.9982 

111.97 

111.77 

111.73 

111.9 

0.05 

(0.9991) 

115.69 

115.59 

115.58 

115.5 

0.02 

(0.9996) 

119.90 

119.85 

119.83 

119.8 

0.01 

(0.9998) 

122.37 

122.35 

122.32 

122.4 

0.005 

(0.9999V 

124.34 

124.33 

124.38 

124.4 

0.002 

1.0000 

126.24 

126.24 

126.31 

126.3 

0.001 

127.27 

127.27 

127.32 

127.2 

0.0005 

128.04 

128.04 

128.05 

127.6 

0.0002 

128.70 

128.70 

128.68 

127.9 

0.0001 

129.00 

129.00 

128.96 

128.1 

0.0 

! 129.50 

129.50 

129.50 

128.3 

216 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVII 


Problem  1. — Calculate  the  specific  conductance  of  0.0075X  KC1  solu- 
tion at  18°  using  (1)  the  Kraus  equation  and  (2)  the  Bates  equation  and  the 
values  of  the  constants  given  in  Table  XXII.  Calculate,  using  Ostwald’s 
dilution  law,  the  specific  conductance  of  0.02A  HC2H3O2  solution  at  18° 
using  the  data  given  in  Table  XX. 

2.  Degree  of  Ionization  and  Conductance  Ratio,  (a)  Ions 
and  Ion-constituents. — Consider  a solution  of  salt  CA,  of  con- 
centration C equivalents  per  liter.  The  equivalent  conduct- 
ance of  this  solution  may  be  regarded  as  made  up  of  two  parts: 

(1)  The  equivalent  conductance  of  the  ion-constituent,  C,  and 

(2)  the  equivalent  conductance  of  the  ion-constituent,  A,  that  is, 

A = Ac+Aa  (2) 

The  distinction  between  ion  and  ion-constituent  should  be  clearly 
understood.  The  equivalent  conductance  (Ac+)  of  C-ion  re- 
fers to  the  conductance  of  one  equivalent  weight  of  the  con- 
stituent.C all  of  which  is  in  the  form  of  ions,  while  the  equivalent 
conductance  (Ac)  of  the  ion-constituent  C refers  to  that  part 
of  the  conductance  which  is  due  to  the  constituent  C,  part  of 
which  may  exist  in  the  solution  in  the  form  of  ions  and  the 
remainder  in  the  condition  of  un-ionized  molecules.  It  is  evident 
from  the  definitions  of  these  two  quantities  that  they  are  related 
to  each  other  by  the  equations 

Ac  = aAc+  (3) 

and  Aa  = a\a-  (4) 

where  Ac  and  A0  are  the  equivalent  conductances  of  the  ion- 
constituents,  C and  A,  in  a solution  of  the  salt  CA,  a is  the  de- 
gree of  ionization  (XV,  4)  of  the  salt  in  this  solution,  and  Ac+ 
and  Aa-  are  the  two  ion-conductances  in  this  same  solution. 
Ac+  and  Aa-  will  in  general  differ  from  A0c+  and  A0a-,  the  ion- 
conductances  at  infinite  dilution,  because  the  ions  in  the  salt 
solution  will  not  in  general  be  able  to  move  with  the  same 
velocity  as  in  pure  water,  since  the  velocity  with  which  ions  move 
through  any  medium  will  depend  upon  the  frictional  resistance 
which  they  experience  in  that  medium  and  this  frictional  re- 
sistance will  in  general  be  different  in  a salt  solution  from  what  it 
is  in  pure  water. 


Sec.  2]  CONDUCTANCE  AND  DEGREE  OF  IONIZATION  217 


(b)  The  Calculation  of  Degree  of  Ionization.— It  may  happen 
with  a given  solution  that  the  resistance  experienced  by  one  of  the 
ion  species  in  moving  through  the  solution  is  practically  the 
same  as  that  which  it  experiences  in  pure  water.  If,  for  example, 
this  were  the  case  for  the  cation,  then  it  is  evident  that  Ac+ 
will  be  practically  equal  to  A0c+  and  equation  (3)  would  become 


Ac 

Aqc+ 


(5) 


and.  it  is  evident  that  by  means  of  this  equation  we  can  calculate 
the  degree  of  ionization  of  the  salt  in  this  solution. 

The  equivalent  conductance  (Ac,  Aa)  of  any  ion-constituent  in 
a solution  is  by  definition  equal  to  the  total  equivalent  con- 
ductance of  the  electrolyte  in  that  solution  multiplied  by  the 
fractional  part  of  the  current-carrying  done  by  the  ion-con- 
stituent in  question  in  that  solution.  This  fraction  is  called 
the  transference  number  ( nc , na)  of  the  ion-constituent  and  the 
methods  by  which  it  is  measured  are  described  in  the  next  chapter. 
We  have,  therefore, 

Ac  = ncA  (6) 


and  hence  equation  (5)  may  be  written 


nc  A 
Aoc+ 


(7) 


If  the  solution  is  so  dilute  that  both  ions  experience  practically 
the  same  resistance  as  they  do  in  pure  water  we  have  (equation  5) 


A c Aa 

Aoc+  Aoa~ 

(8) 

and  also  by  the  principles  of  proportion 

Ac  “f~  Aa 

(9) 

Aoc+  + A()a- 

or 

A 

(10) 

OL  = , 

Ao 

that  is,  under  these  circumstances  the  degree  of  ionization  of  a 
uni-univalent  electrolyte  at  concentration  C is  equal  to  the  ratio 
of  its  equivalent  conductance  at  this  concentration,  to  its  equiva- 


218 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVII 


lent  conductance  at  infinite  dilution.  Equation  (10)  is  the  one 
most  commonly  employed  for  calculating  degree  of  ionization 
from  conductance  data  and  is  the  one  which  we  shall  employ 
throughout  this  book  except  where  otherwise  indicated.  It  is 
evident  from  its  derivation  that  the  equation  is  only  valid  for  those 
solutions  in  which  both  ions  are  able  to  move  with  the  same  speed  as 
they  do  at  infinite  dilution. 

(c)  The  Viscosity  Correction. — In  cases  where  the  above  con- 
dition is  not  fulfilled  attempts  have  been  made  to  correct  for  the 
effect  of  frictional  resistance,  on  the  assumption  that  Stokes’ 
Law  (IX,  3)  is  obeyed  and  hence  that  the  equivalent  conductance 
of  an  ion  is  inversely  proportional  to  the  viscosity  (III,  6)  of  the 
medium  through  which  it  moves.  This  assumption  leads  to  the 
equations 


and 


Aoc+  Vo  Ao<j-  yo 


(11) 


A T) 

Ao  770 


(12) 


corresponding  to  equations  (8)  and  (10).  The  assumption  is  of 
doubtful  validity,  however,  and  the  whole  question  of  the 
proper  method  of  correcting  for  the  influence  of  viscosity  is  in  a 
very  unsatisfactory  condition.  Where  a correction  for  viscosity 
is  necessary  equations  (11)  and  (12)  may  perhaps  be  employed  as 
first  approximations  in  many  cases  with  a fair  degree  of  accuracy, 
and  this  is  about  all  that  we  can  safely  say  at  present. 

Some  prominent  investigators  in  this  field  even  incline  to  the 
opinion  that  the  speeds  of  the  ions  actually  increase  with  increas- 
ing concentration  of  the  electrolyte,  and  that  even  in  solutions  so 
dilute  that  the  viscosity  of  the  solution  is  practically  identical 
with  that  of  pure  water  the  ions  move  with  decidedly  greater 
velocities  than  they  do  at  infinite  dilution.  If  such  is  the  case, 
the  degrees  of  ionization  calculated  from  the  conductance  ratio 
are  all  too  high.  The  evidence  at  present  (1915)  available  on  this 
point  is  not  very  conclusive  either  way,  but  in  the  treatment  of 
the  subject  in  the  following  pages  we  shall  in  accordance  with 
the  general  practice  take  the  conductance  ratio  as  a measure  of 
the  quantity  a,  and  shall  attempt  to  interpret  apparent  anomalies 
by  assigning  causes  other  than  an  erroneous  determination  of  a. 


Sec.  3]  CONDUCTANCE  AND  DEGREE  OF  IONIZATION  219 


* 


< 


Problem  3. — If  in  a solution  containing  0.05  mole  each  of  KC1  and  LiCl 
each  salt  is  85  per  cent,  ionized,  what  would  be  the  specific  conductance  of 
the  solution  (1)  at  18°  and  (2)  at  25°?  Use  Table  XXI. 


3.  Degree  of  Ionization  and  Type  of  Electrolyte. — Salts 
belonging  to  the  same  ionic  type  (XV,  3b)  have  at  the  same 
equivalent  concentrations  approximately  the  same  value  of  the 


conductance  ratio, 


A 

Ao 


Table  XXIII  shows  some  average  values 


of  this  ratio  for  the  three  simplest  types. 


Table  XXIII 

Values  of  -7-  at  different  concentrations  for  three  types  of  electrolytes 
Ao 


Type 

Example 

0.001 

0.01 

0.05 

0 . 1 Normal 

Uni-univalent 

KN03 

0.96 

0.92 

0.87 

0.84 

Uni-bivalent 

f BaCl  2 1 
;k2SO,; 

0.94 

0.87 

0.78 

0.73 

Bi-bivalent 

MgS04 

0.86 

0.64 

0.47 

0.41 

For  uni-univalent  and  probably  also  for  bi-bivalent  salts 
these  values  represent,  according  to  equation  (10),  the  degrees 
of  ionization  of  the  salts,  because  salts  of  these  types  ionize 
directly  into  the  ions  to  which  Ao  corresponds.  Such  is  not 
the  case,  however,  with  the  uni-bivalent,  the  uni-trivalent  and 
the  bi-trivalent  types  because  they  ionize  in  such  a way  as  to 
give  intermediate  ions.  Thus  K2SO4  ionizes  so  as  to  give  the 
ions  K+,  S04~~,  and  KS04~.  At  infinite  dilution  the  inter- 
mediate ions  KS04“  are  all  broken  up  into  K+  and  S04  ions 
so  that  the  Ao  value  corresponds  only  to  these  two  ion  species, 
but  at  any  finite  concentration  the  A value  is  made  up  of  the 
separate  conductances  of  three  species  of  ions,  the  relative 
amounts  of  which  are  not  known.  Owing  to  the  presence  of  these 
intermediate  ions  equation  (10)  cannot  be  employed  to  calculate 
a for  such  electrolytes.  We  have  no  very  satisfactory  method 
for  determining  the  degree  of  ionization  and  the  ion  concentra- 
tions for  salts  like  BaCl2,  although  it  can  be  done  approximately 
in  some  instances  by  a combination  of  several  methods. 

We  can,  however,  make  the  general  statement  that  all  salts 
are  highly  ionized  in  dilute  aqueous  solution.  There  are  a few 


220 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVII 


exceptions  to  this  rule  but  they  are  very  few  and  the  rule  is  one 
which  the  student  should  remember.  Some  of  the  exceptions 
are  lead  acetate,  ferric  sulphocyanate,  cadmium  chloride  (a  — 
45  per  cent,  at  0.1  N)  and  the  mercuric  halides  (a  = 0.1  per 
cent,  at  0.1  N). 

Acids  and  bases,  unlike  salts,  exhibit  at  any  moderate  con- 
centration, such  as  0.1  normal,  every  possible  degree  of  ioniza- 
tion between  a small  fraction  of  1 per  cent,  up  to  nearly  90 
per  cent,  according  to  the  nature  of  the  substance.  There  is, 
to  be  sure,  a fairly  large  group  of  uni-univalent  acids  and  bases, 
including  HC1,  HBr,  HI,  HN03,  HC103,  KOH,  NaOH,  LiOH, 
which  have  ionization  values  comparable  with  those  of  the  uni- 
univalent salts,  with  which,  therefore,  they  may  be  classed; 
and  likewise  the  uni-bivalent  bases  of  the  alkaline  earths  are 
also  highly  ionized  in  dilute  solution,  although  perhaps  somewhat 
less  so  than  those  of  the  alkali  metals.  But  outside  of  these 
groups  all  possible  values  of  a are  met  with,  as  illustrated  by 
the  following  values  of  the  percentage  ionization  (100a)  at  25° 
and  0.1  normal:  NH4OH,  1.3  per  cent.;  H2S03,  34  per  cent, 
(into  H+  and  HS03~);  H3P04,  28  per  cent,  (into  H+  and 
H2P04“);  HN02,  7 per  cent.;  HC2H302,  1.3  per  cent.;  H2C03, 
H2S,  HCIO,  HCN,  HB02,  all  less  than  0.2  per  cent. 

Polybasic  acids  are  known  to  ionize  in  stages,  giving  rise  to 
the  intermediate  ion;  and  the  first  hydrogen  is  almost  always 
much  more  dissociated  than  the  second,  and  the  second  much 
more  than  the  third.  Thus  H2S03  at  0.1  normal  at  25°  is  about 
34  per  cent,  dissociated  into  H+  and  HS03“,  and  less  than  0.01 
per  cent,  dissociated  into  H+ and  S03  . 

4.  Comparison  of  Degrees  of  Ionization  Calculated  by  the 
Freezing-point  and  Conductance  Methods. — As  explained  above 
(XV,  4),  the  freezing-point  method  can  frequently  be  employed 
to  calculate  the  approximate  degree  of  ionization  of  uni-univalent 
electrolytes  owing  to  a partial  compensation  of  sources  of  error. 
A comparison  of  the  values  obtained  by  the  two  methods  in  the 
case  of  CsN03,  KC1  and  LiCl  is  shown  in  Table  XXIV.  In 
the  case  of  CsN03  the  values  obtained  from  the  freezing-point 
method  are  all  too  small.  This  may,  at  least  partially,  be 
ascribed  to  the  powerful  effect  which  the  ions  have  upon  the 
thermodynamic  environment  in  the  solution.  In  the  case 


Sec.  4]  CONDUCTANCE  AND  DEGREE  OF  IONIZATION  221 


of  LiCl  the  values  obtained  by  the  freezing-point  method  are  all 
too  large.  This  is  due  largely  to  the  fact  that  LiCl  is  highly 
hydrated  in  solution  and  no  account  was  taken  of  this  in  the 
calculation.  In  the  case  of  KC1  these  two  sources  of  error 
appear  to  compensate  each  other  almost  exactly  and  the  values 
of  a by  the  two  methods  agree  within  the  experimental  errors 
of  the  measurements.  LiCl  and  CsN03  are  extreme  cases. 
Practically  all  other  uni-univalent  salts  lie  within  the  limits 
set  by  these  two.  It  is  evident  from  these  results  that  agree- 
ment between  the  values  obtained  by  the  two  methods  is  no 
evidence  that  the  values  are  correct  (as  has  been  concluded  by 
some  investigators),  for  the  agreement  can  only  be  an  accidental 
one.  Even  the  accidental  very  close  agreement  in  the  case  of 
KC1  at  0°  disappears  at  25°  when  vapor  pressure  measurements 
are  made  the  basis  for  calculating  a (Cf.  XV,  2 and  problem  4). 

Table  XXIV 

Comparison  of  the  degrees  of  ionization  calculated  by  the  freezing-point 
method,  equation  (7,  XIV),  and  by  the  conductance  method,  equation  (12 
XVII).  “p,  by  the  freezing-point  method.  by  the  conductance 

method.  [Cf.  Jour.  Amer.  Chem.  Soc.,  33,  1702  (1910)]. 


A tp 

(l  + aF)i!^i 

100«f 

j 100  a A 

CsN03 

0.025 

0.086° 

0.046s 

85. 

89.1 

0.10 

0.325 

0.175 

75. 

80.8 

0.20 

0.622 

0.3357 

68. 

74.6 

0.50 

1.419 

0.770 

54. 

64.5 

KC1 

0.025 

0.089 

0.9479 

92. 

93.1 

0.10 

0.345 

0.186 

86. 

87.9 

0.20 

0.680 

0.367 

84. 

84.7 

0.50 

1.658 

0.899 

79.8 

79.6 

LiCl 

0.025 

0.090 

0 . 0485 

94. 

90.7 

0.10 

0.351 

0.189 

89. 

85.7 

0.20 

0.694 

0.374 

87. 

82.3 

0.50 

1.791 

0.924 

84.8 

78.0 

5.  Degree  of  Ionization  and  Temperature. — At  room  tempera- 
tures the  value  of  a for  uni-univalent  strong  electrolytes  changes 
with  the  temperature  at  a rate  of  less  than  0.1  per  cent,  per  de- 


222 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVII 


gree.  For  KC1  at  about  0.08  normal,  A.  A.  Noyes  found  for 
100a  the  following  values: 

0°  18°  100°  156°  218°  281°  306° 

89.3  87.3  82.6  79.7  77.3  72  64 

which  are  typical  of  most  salts  of  this  type.  In  the  case  of  weak 
electrolytes  no  general  rule  as  to  direction  and  magnitude  can  be 
given,  although,  as  we  shall  see  later  (XXII,  10),  the  rate  of 
change  with  the  temperature  can  be  calculated  from  the  heat  of 
ionization  of  the  electrolyte. 


CHAPTER  XVIII 


ELECTRICAL  TRANSFERENCE 

1.  The  Phenomenon  of  Electrical  Transference. — We  have 
seen  (XVI,  3)  that  when  a current  of  electricity  is  passed  through 
a solution  of  an  electrolyte  all  of  the  anions  move  toward  the 
anode  and  all  of  the  cations  toward  the  cathode.  Since  the 
different  ions  move  with  different  speeds,  the  faster  moving 
ones  will,  other  things  being  equal  (XVI,  3),  do  more  of  the 
current  carrying  than  the  slower  moving  ones.  Through  any 
cross  section  of  the  solution,  however,  the  number  of  equivalents 
of  cation  which  move  toward  the  cathode  plus  the  number  of 
equivalents  of  anion  which  move  toward  the  anode  must  by 
definition  be  equal  to  the  total  number  of  equivalents  of  elec- 
tricity (XVI,  5)  passed  through  the  solution,  or 

Nc+Na  = Ne 

and  the  ratios 

Nc  _NC_ 

Nc+Na  Ne 

and 

Na  _Na_ 

Nc+Na  Ne  n“ 

evidently  represent  the  fractional  part  of  the  total  current 
carried  by  each  species  of  ion  and  also  the  number  of  equivalents 
of  each  ion  species  as  well  as  the  number  of  equivalents  of  each 
ion-constituent  transferred  in  the  one  direction  or  the  other' 
through  the  solution  during  the  passage  of  one  faraday  of  elec- 
tricity. The  quantities  nc  and  na  are  called  the  transference 
numbers  (XVII,  2b)  of  the  ions  and  of  the  ion-constituents  indi- 
cated by  the  subscripts.  As  a result  of  this  transfer  of  ions 
through  the  solution,  concentration  changes  take  place  around 
the  electrodes.  The  nature  of  these  changes  and  their  relation 
to  the  transference  number  may  be  best  understood  by  the  con- 
sideration of  a Hittorf  transference  experiment,  \[]  | 

|223 


(1) 

(2) 

(3) 


224 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVIII 


2.  A Hittorf  Transference  Experiment.— Let  a solution  of 

silver  nitrate  be  electrolyzed  between  silver  electrodes  in  the 
apparatus  shown  diagrammatically  in  Fig.  31.  At  the  end  of  the 
electrolysis  let  the  solution  be  drawn  off  in  five  separate  portions, 
three  middle  portions  and  two  electrode  portions,  as  indicated 

in  the  figure,  precautions  be- 
ing taken  to  prevent  mixing 
during  this  operation.  Let 
each  portion  be  weighed  and 
analyzed  to  determine  the  per 
cent,  of  water  and  the  per 
cent,  of  silver  nitrate  which  it 
contains.  The  result  of  these 
analyses  will  show  that  the 
three  middle  portions  have 
the  same  composition  as  the 
original  solution  (if  the  ex- 
periment has  been  properly 
conducted)  but  that  the  con- 
centration of  silver  nitrate  has 
increased  in  the  anode  portion 
and  decreased  in  the  cathode 
portion.  The  mechanism  of 
these  concentration  changes  is 
the  following: 

At  the  anode,  in  accordance 
with  Faradays  law,  Ne  equiva- 
lents of  silver  have  dissolved 
from  the  electrode  and  passed 
into  solution  as  silver  ion 


Fig.  31. — Diagrammatic  representa- 
tion of  a transference  apparatus  show- 
ing division  into  portions  for  analysis. 


(XVI,  4a),  Ne  being  the  number  of  equivalents  of  electricity 
passed  through  the  solution  during  the  experiment.  At  the 
same  time  a certain  amount,  Nc  equivalents,  of  silver  ion 
has  moved  out  of  the  anode  portion,  and  Ne  — Nc  equivalents  of 
nitrate  ion  have  moved  into  the  anode  portion.  (See  equation 
(1).)  Suppose  that  the  analysis  of  the  anode  portion  gives  the 
composition,  N2  equivalents  of  AgN03  in  mw  grams  of  water. 
The  original  solution  before  electrolysis  contained  Ni  equivalents 
of  AgN03  in  mw  grams  of  water.  The  increase , N2  — Ni,  must 


Sec.  3] 


ELECTRICAL  TRANSFERENCE 


225 


evidently  be  equal  to  Ne  — Nc,  that  is,  it  must  be  equal  to  the 
silver  which  has  come  into  the  anode  portion  from  the  electrode 
diminished  by  the  silver  which  has  gone  out  of  the  anode  portion 
owing  to  the  migration  of  silver  ions  toward  the  cathode.  Hence 
by  definition,  equation  (2),  the  transference  number  of  the  silver 
ion  is  evidently 


C Nx-Nt  + N, 

Ne  N 


and  that  of  the  nitrate  ion  is  by  definition  (equations  2 and  3), 
na  = 1 — nc.  At  the  cathode  the  reverse  of  the  above  process 
takes  place  and  from  the  analysis  of  the  cathode  portion  the 
two  transference  numbers  can  also  be  calculated  by  means  of 
equation  (4),  thus  giving  a valuable  check  on  the  accuracy  of 
the  experimental  work. 

In  employing  equation  (4)  the  following  points  which  follow 
from  its  derivation,  should  be  kept  clearly  in  mind:  From  the 
total  mass  and  composition  of  the  electrode  portion  as  calcu- 
lated from  the  results  of  the  analysis  and  the  knowledge  of  the 
electrode  process,  the  mass  of  water,  mw,  in  the  electrode  portion 
can  be  computed.  The  analysis  also  gives  the  number  of 
equivalents,  N 2,  of  the  ion-constituent  whose  transference 
number  is  desired,  which  are  present  in  the  mw  grams  of  water 
in  the  electrode  portion.  From  the  known  composition  of  the 
original  solution  we  can  calculate  the  number  of  equivalents, 
Ni,  of  this  ion-constituent  which  were  present  in  mw  grams  of 
water  in  the  original  solution.  Ne  in  the  numerator  of  equation 
(4)  is  the  number  of  equivalents  of  the  ion-constituent  in  ques- 
tion which  have  passed  into  the  electrode  portion  from  the 
electrode.  If  the  ion-constituent  happens  to  have  gone  out  of 
the  solution  on  to  the  electrode,  then  Ne  is  obviously  a negative 
quantity.  In  many  cases  it  will  be  zero.  Ne  in  the  denominator 
of  equation  (4)  is  the  number  of  equivalents  of  electricity  passed 
through  the  solution  during  the  experiment,  as  measured  by  a 
suitable  coulometer  (XVI,  5)  connected  in  series  with  the  trans- 
ference apparatus. 

3.  True  Transference  Numbers  and  Ionic  Hydration. — It 

will  be  noted  that  the  above  calculation  of  transference  numbers 
by  the  Hittorf  method  assumes  that  all  of  the  water  in  the  solu- 


226 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVIII 


tion  remains  stationary  during  the  passage  of  the  current,  and 
the  change  in  composition  which  occurs  in  the  neighborhood 
of  the  electrode  is  calculated  with  reference  to  the  water,  assumed 
to  be  stationary.  The  assumption  is  not  strictly  true,  since  the 
ions  are  hydrated  (XV,  1)  and  hence  some  water  is  also  carried 
with  them  as  they  move  through  the  solution.  In  dilute  solu- 
tion the  water  so  carried  is  negligible  but  in  concentrated  solu- 
tion it  is  not  and  it  becomes  necessary  to  add  to  the  solution  a 
third  substance,  such  as  sugar,  which  will  remain  stationary 
during  the  passage  of  the  current  and  which  can  serve  as  the 
reference  substance  for  measuring  the  concentration  changes 
of  the  salt  and  the  water.  In  this  way  the  comparative  degrees 
of  hydration  of  different  ions  have  been  measured. 

Table  XXV 

Ionic  hydrations  and  “true”  transference  numbers  at  25°  obtained  from 
transference  experiments  in  the  presence  of  a non-electrolyte  as  a refer- 
ence substance.  ncT  = true  transference  number  of  the  cation,  n oo  = 
transference  number  at  infinite  dilution,  ncH  = Hittorf  transference  number 
at  1.3V,  Nw  = number  of  water  molecules  attached  to  an  ion.  (See  Jour. 
Amer.  Chem.  Soc.  37,  000  (1915)). 


Electrolyte 
(Cone.  1.3  V) 

A nw/ncT 

An. 

ncT 

1 n oo 

ncH 

HC1 

0.28  + 0.04 

0.24  + 0.04 

0.844 

0.847 

0.82 

CsCl 

0.67+0.1 

0.33+0.06 

0.491 

0.491 

0.485 

KC1 

1.3  ±0.2 

0.60+0.08 

0.495 

0.495 

0.482 

NaCl 

2.0  ±0.2 

0.76+0.08 

0.383 

0.396 

0.366 

LiCl 

4.7  ±0.4 

1.5  ±0.1 

0.304 

0 . 330 

0.278 

nJ1 

= 0. 

28 

1+0. 

04 

c+0. 185V.C1 

(1) 

NWC* 

= 0. 

.67+0, 

1 

+ 1.03 

Nwcl 

(2) 

NWK 

= 1. 

3 

±0. 

.2 

+ 1.02 

Nwcl 

(3) 

JV„Na 

= 2. 

0 

±0. 

2 

+ 1.61 

Nwcl 

(4) 

Nwu 

= 4 

.7 

±0 

.4 

+2.29 

Nwcl 

(5) 

Transference  numbers  obtained  by  using  a reference  sub- 
stance which  remains  stationary  during  the  passage  of  the 
current  are  called  “true”  transference  numbers,  while  those  ob- 
tained by  the  use  of  water  as  the  reference  substance  are  called 
“Hittorf”  transference  numbers.  In  dilute  solution,  however, 


Sec.  5] 


ELECTRICAL  TRANSFERENCE 


227 


the  two  become  practically  identical  as  is  evident  from  the 
following  relation  which  connects  them 


na  = na 


Ns _ 

’Nu 


(5) 


naT  and  naH  are  respectively  the  true  and  the  Hittorf  trans- 
ference numbers,  A nw  is  the  number  of  moles  of  H20  trans- 
ferred from  anode  to  cathode  per  faraday  of  electricity  and 

N3  . , . . . electrolyte 

7r  is  the  molal  ratio,  . 

N w tl2U 

relative  ionic  hydrations  and  true  transference  numbers  obtained 

in  this  way  in  the  case  of  the  uni-univalent  chlorides  are  shown 

in  Table  XXV. 


-,  in  the  original  solution.  The 


Note. — Compare  the  relative  degrees  of  hydration  of  the  alkali  ions  as 
shown  in  Table  XXV  with  their  ion-conductances  given  in  Table  XXI  and 
with  the  atomic  weights  of  the  elements,  Table  I. 

4.  Change  of  Transference  Numbers  with  Concentration  and 
Temperature. — In  the  case  of  uni-univalent  salts  the  trans- 
ference numbers  change  very  little  with  the  concentration  as 
long  as  the  latter  does  not  exceed  a moderate  value,  say  0.01 
normal.  This  is  illustrated  by  the  following  values  of  ncn 
at  18°. 

Table  XXVI 


Values  of  ncH  at  18° 


C 

0.0  0.005 

0.01 

0.02 

0.05 

0.1 

0.2 

0.3 

0.5 

1.0 

KC1 

. . 0.496  0. 496 

0.496 

0.496 

0.496 

0.495 

0.494 

NaCl 

. . 0.3980.396 

0.396 

0.396 

0.395 

0.393 

0.390 

0.388 

0.382 

0.369 

Li  Cl 

. . 0.338  

0.332 

0.328 

0.320 

0.313 

0.304 

0.299 

With  rising  temperature  all  transference  numbers  which  have 
been  measured  exhibit  a tendency  to  approach  0.5. 

For  compiled  tables  of  the  best  transference  data  see  reference 

Problem  1. — If  the  anion  transference  number  of  potassium  chloride  is 
0.504  and  if  the  equivalent  conductance  at  infinite  dilution  has  the  following 
values:  129.5  for  KC1,  125.9  for  KNO3,  and  104.9  for  NaNOp,  what  is  the 
transference  number  of  sodium  nitrate? 

5.  Transference  and  Ion  Mobility. — A charged  body  in  an 
electric  field  is  acted  upon  by  a force,  /,  equal  to  the  product  of 
the  charge,  q,  into  the  potential  gradient,  (dE/dl)  at  that  point 
in  the  field,  or 


f=q(dE/dl) 


(6) 


228 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVIII 


where  d E is  the  change  in  potential  in  the  distance  d l.  More- 
over, the  velocity  of  a small  body  moving  through  a medium  of 
great  frictional  resistance  is  proportional  to  the  force  acting  upon 
it.  Hence  the  velocity,  u,  with  which  an  ion  moves  through  a 
solution  of  a given  vicosity  is  proportional  to  the  potential  gra- 
dient prevailing  within  the  solution  ( q being  obviously  a constant 
for  a given  ion),  that  is, 

uc+=  Uc+  (dE/dl)  and  ua-=  Ua-  (dE/dl)  (7) 

where  the  proportionality  factor  ( Uc+ , Ua- ) is  obviously  the 
velocity  of  the  ion  under  unit  potential  gradient.  It  is  called 
the  mobility  of  the  ion  and  is  a characteristic  property  which 
varies  with  the  temperature  and  with  the  nature  of  the  medium 
through  which  the  ion  moves.  Similarly  the  velocity  with  which 
a given  ion-constituent  is  transferred  through  the  solution  will 
be 

uc  = aUc+  (dE/dl)  = Uc(dE/dl)  (8) 

and  similarly  for  the  anion. 


If  a constant  current,  I,  of  electricity  be  passed  for  t seconds 
through  a solution  containing  C equivalents  of  a uni-univalent 
electrolyte,  CA,  per  liter,  placed  in  a cylindrical  tube  of  cross 
section  A,  between  parallel  electrodes  1 cm.  apart,  a consideration 
of  Fig.  32,  will  show  that  the  number  of  equivalents,  Nc,  of  the 
cation  constituent  which  pass  through  any  cross  section  of  the 
tube  in  this  time,  will  be 

Nc  = uctA  (0.001C)  (9) 

or,  since  uc=  Uc  (dE/dl),  by  equation  (8),  and  dE/dl  equals 
E/l  for  a constant  current  in  a homogeneous  conductor  of  uni- 
form cross  section,  (Prove  this,  see  equations  (25)  and  (27), 
XIV)  equation  (9)  becomes 

Nc=Uc(E/l)  ^(O.OOIC) 


(10) 


Sec.  6] 


ELECTRICAL  TRANSFERENCE 


229 


and  similarly  for  the  number  of  equivalents  of  anion  constituent 
passing  in  the  opposite  direction, 

Na  = Ua(E/l)tA(OmiC)  (11) 


If  we  divide  each  of  these  equations  by  their  sum  we  obtain  an 
expression  for  each  of  the  transference  numbers  (see  equations 
(2)  and  (3))  in  terms  of  the  mobilities  of  the  ion-constituents. 


nc 


Uc 

Uc-\-  U a 


and  na  — 


Ue~\~  Ua 


(12) 


6.  Transference  and  Ion-conductances. — In  the  example  just 
considered  (Fig.  32),  the  total  number  of  coulombs,  q,  of  elec- 
tricity passed  through  our  solution  in  the  time  t is  by  definition 
(equation  (24),  XVI) 


q = It  (13) 

By  Faraday’s  law  (equation  (23),  XVI),  it  is  also 


q=FNe  (14) 

and  Ne  the  number  of  equivalents  of  electricity  passed  is,  ac- 
cording to  equation  (1),  expressed  by  the  sum  of  equations  (10) 
and  (11),  thus  giving  us 


It  = q=FNe=F(Nc+Na)=F(Uc+Ua)  (E/l)tA(0.001C)  (15) 

By  simply  combining  this  equation  algebraically  with  equations 
(25,  26,  27,  and  29,  XVI),  (prove  this),  we  obtain  the  important 
relation, 

A = F(Uc+Ua)=FUc+FUa  (16) 

and  hence 

A =FoiUc+-\-FaiUc-  (17) 

or  in  words,  the  equivalent  conductance  of  a uni-univalent  elec- 
trolyte is  equal  to  the  faraday  multiplied  either  by  the  sum  of 
the  mobilities  of  its  ion-constituents  or  by  the  product  of  the 
degree  of  ionization  into  the  sum  of  the  mobilities  of  its  ions. 
But  the  equivalent  conductance  of  the  salt  is  equal  to  the  sum 
of  the  equivalent  conductances  of  its  two  ion-constituents,  or 
to  the  product  of  its  degree  of  ionization  into  the  sum  of  the 


230 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVIII 


conductances  of  its  two  ions  (XVII,  2,  equations  2,  3,  and  4), 
that  is, 

A = AC  + Aa  (18) 

and 

A = a(Ac++Aa-)  (19) 

and  by  comparing  these  two  equations  with  equations  (16)  and 
(17),  it  is  evident  that 

A e=FUc,  A a=FUa  (20) 

and  AC+=FUC+-,  Aa+=FUa+  (21) 

and  the  absolute  velocity  in  centimeters  per  second  with  which  a 
given  ion  (or  ion-constituent)  moves  through  a solution  under 
unit  potential  gradient  is  evidently  equal  to  the  equivalent 
conductance  of  the  ion  (or  ion-constituent)  in  the  solution  in 
question,  divided  by  the  faraday. 

Equations  (12)  and  (16)  combined  with  (20)  and  (21)  give 

Ac=Anc,  Aa  = Ana  (22) 

and  Ac+  = aAnc,  Aa-  = aAna  (23) 

both  of  which  at  infinite  dilution  become  (since  a = 1) 

Aoc  A0<+  A0tiC}  and  Aoa==  Aoa-  =AQ7ia  (24) 

We  have  seen  how  values  of  A0  and  of  n are  obtained  experi- 
mentally (XVII,  1 and  XVIII,  2 and  7)  and  it  is  clear  that  by 
means  of  equation  (24)  the  individual  ion-conductances  at  in- 
finite dilution  recorded  in  Table  XXI  can  be  computed.  To 
compute  such  a table  all  that  is  necessary  is  a set  of  A0  values  for  a 
sufficient  number  of  electrolytes  to  include  all  of  the  desired  ion 
species  and  the  transference  number  of  a single  ion  species  of 
one  of  those  electrolytes  in  very  dilute  solution.  The  values 
givemin  Table  XXI  are  based  upon  the  transference  number  of 
potassium  chloride  (see  Table  XXVI). 

Problem  2. — A transference  experiment  is  made  with  a solution  of  silver 
nitrate  (0.00739  gram  of  silver  nitrate  per  gram  of  water)  using  two  silver 
electrodes.  A silver  coulometer  in  the  circuit  shows  a deposit  of  0.0780 
gram  of  silver.  At  the  end  of  the  experiment  the  anode  portion  weighing 
23.38  grams  is  removed  and  found  on  analysis  to  contain  0.2361  gram  of 
silver  nitrate.  Calculate  the  transference  numbers  of  the  silver  and  nitrate 
ions  in  a silver  nitrate  solution  of  this  concentration.  The  cathode  portion 
wreighs  25.00  grams.  How  much  silver  nitrate  does  it  contain? 


Sec.  7] 


ELECTRICAL  TRANSFERENCE 


231 


A) 


C A 


7.  Determination  of  Transference  Numbers  by  the  Moving 
Boundary  Method. — From  equations  (7)  and  (12)  it  is  evident 
that  the  ratio  of  the  transference  numbers  of  the  two  ions  in  a 
given  salt  solution  is  equal  to  the  ratio  of  the  velocities  with  which 
the  two  ion-constituents  move  through  the  solution 
and  hence  equal  to  the  ratio  of  the  distances  covered 
by  the  two  ion-constituents  in  a given  time  during  the 
passage  of  the  current.  These  two  distances  can  be 
directly  measured  with  the  apparatus  shown  dia- 
gramatically  in  Fig.  33.  A solution  of  the  salt,  CA, 
under  investigation  is  placed  between  one  of  the  salt 
CA'  and  another  of  the  salt  C'A,  having  respectively 
the  same  cation  and  anion  as  the  salt  CA.  The 
boundaries  aa  and  bb  separating  the  solutions  are 
easily  visible  because  of  the  different  refractive  indices 
(VIII,  3)  of  the  solutions.  A current  is  passed  through 
the  apparatus  in  the  direction  indicated  by  the  short 
arrows.  Boundary  aa  moves  upward  to  a' a'  and  evi- 
dently represents  the  distance  covered  by  the  anion- 
constituent  A during  the  passage  of  the  current. 

Similarly  boundary  bb  moves  downward  to  b'b'  and 
represents  the  distance  covered  by  the  cation-constitu- 
ent. The  ratio  of  the  two  distances  is  the  ratio  of 
the  two  transference  numbers. 


CA 


CA 


Fig.  33. 


Problem  3. — Prove  that  the  two  boundaries  will  remain  sharp 
while  the  current  passes,  if  Uaf  < Ua  and  Vc'  < Uc.  The  salts 
are  so  chosen  that  these  conditions  are  fulfilled.  If  one  desired 
to  measure  the  transference  number  of  NaCl  by  this  method,  what  salts 
might  be  used  for  and  C'A  and  CA'? 


^ REVIEW  PROBLEMS 

The  atomic  weight  table  and  the  values  of  general  constants  such  as  the 
faraday  may  be  employed  in  solving  the  following  problems  but  no  other 
data  except  those  given  with  the  problem  are  to  be  used.  The  influence  of 
viscosity  may  be  neglected  in  computing  a. 

Problem  4. — Using  a silver  anode  and  a silver  chloride  cathode,  0.04974 
equivalent  of  electricity  was  passed  through  an  aqueous  solution  containing 
8.108  per  cent,  of  KC1  and  4.418  per  cent,  of  raffinose  contained  in  a suitable 
1 transference  apparatus.  At  the  end  of  the  experiment  the  anode  portion 

weighing  103.21  grams  was  analyzed  and  found  to  contain  6.510  per  cent. 


232 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XVIII 


of  KC1  and  4.516  per  cent,  of  raffinose.  The  cathode  portion  weighing 
85.28  grams  contained  10.030  per  cent,  of  KC1  and  4.290  per  cent,  of  raffinose. 
Three  middle  portions  were  found  to  have  the  same  composition  as  the 
original  solution.  On  the  assumption  that  the  raffinose  remains  stationary 
during  the  passage  of  the  current  calculate  the  true  transference  number  of 
the  potassium  ion  and  the  amount  of  water  transferred  per  equivalent  of 
electricity.  Assuming  that  NwCl  molecules  of  water  are  attached  to  the 
chloride  ion  calculate  the  number  attached  to  the  potassium  ion.  (Cf. 
Table  XXV.) 

Problem  5. — A solution  containing  0.1  equivalent  of  KC1  per  1000  grams 
.of  H20  has  a specific  resistance  of  89.37  ohms  at  18°  and  is  86.15  per  cent, 
ionized.  This  solution  is  electrolized  in  a transference  apparatus  with  a 
silver  anode  and  a silver  chloride  cathode.  At  the  end  of  the  electrolysis 
the  solution  around  each  electrode  is  neutral  and  no  gas  has  been  evolved  at 
either  electrode.  The  anode  is  carefully  washed,  dried  and  weighed  and  is 
found  to  have  increased  in  weight  by  0.01300  gram.  A cathode  portion  of 
100  grams  is  drawn  off  from  the  neighborhood  of  the  cathode  and  is  found 
on  analysis  to  contain  0.7539  gram  of  KC1.  Calculate  from  these  data 
(a)  the  equivalent  conductance  of  potassium  ion  at  infinite  dilution,  (b) 
the  absolute  velocity  in  centimeters  per  second  of  chloride  ion  in  dilute 
solution  at  18°  when  moving  under  a potential  gradient  of  2 volts  per 
centimeter,  and  (c)  the  absolute  velocity  of  chloride  ion-constituent  under 
the  same  conditions. 

Problem  6. — The  specific  conductance  of  0.01  normal  potassium  nitrate 
solution  at  18°  is  0.001182  mho.  Its  equivalent  conductance  at  infinite 
dilution  at  the  same  temperature  is  125.9  mhos.  Calculate  its  degree  of 
ionization.  On  the  assumption  that  it  has  the  same  degree  of  ionization  at 
0°,  calculate  its  freezing  point. 

Problem  7. — The  equivalent  conductance  of  silver  nitrate  at  infinite 
dilution  is  115.7  mhos  at  18°.  From  the  result  obtained  in  problem  2, 
calculate  the  equivalent  conductances  of  the  silver  and  nitrate  ions  at 
infinite  dilution. 

Problem  8. — From  the  result  obtained  in  problem  7,  together  with  the 
data  given  in  problem  6,  calculate  the  equivalent  conductance  of  infinitely 
dilute  potassium  chloride  solution  at  18°,  the  equivalent  conductance  of 
silver  chloride  at  infinite  dilution  being  119.3  mhos. 

Problem  9. — From  the  result  obtained  in  problem  8,  calculate  approxi- 
mately the  specific  conductance  of  0.05  normal  potassium  chloride  solution 
at  18°.  The  freezing  point  of  this  solution  is  —0.175°.  Assume  the  same 
degree  of  ionization  at  0°  and  18°. 

Problem  10. — Calculate  approximately  the  conductance  of  CT.l  normal 
sodium  nitrate  solution  at  18°  in  the  cell  of  problem  11.  The  freezing  point 
of  the  solution  is  —0.338°  and  its  equivalent  conductance  (18°)  at  infinite 
dilution  is  104.9  mhos. 

Problem  11. — A cylindrical  conductivity  cell  of  1 cm.  radius  is  fitted  tightly 
with  parallel  silver  electrodes  2 cm.  apart  and  is  filled  with  a 0.1  A solution 


Sec.  7] 


ELECTRICAL  TRANSFERENCE 


233 


of  AgNC>3  for  which  the  transference  number  of  the  positive  ion  is  0.468. 
A potential  difference  oi  6.750  volts  causes  a current  of  0.1  ampere  to  flow 
through  the  cell.  The  equivalent  conductance  of  KNO3  at  infinite  dilution 
is  125.9  mhos  and  the  transference  number  of  the  positive  ion  is  0.511. 
Calculate  the  degree  of  ionization  of  AgNC>3  in  0.1  A solution. 

Problem  12. — Calculate  the  degree  of  ionization  of  ammonium  hydroxide 
in  0.1  normal  solution  from  the  following  data:  its  equivalent  conductance 
at  25°  in  0.1  normal  solution  is  4;  the  equivalent  conductance  of  ammonium 
chloride  at  infinite  dilution  is  155,  the  transference  number  for  the  positive 
ion  is  0.50  in  the  case  of  ammonium  chloride  and  0.27  in  the  case  of  am- 
monium hydroxide. 

Problem  13.: — Describe  with  sketch  of  apparatus  an  experiment  by  means 
of  which  one  could  determine,  more  definitely  than  from  the  data  in  problem 
5,  XV,  the  nature  of  the  complex  formed  between  NH.,  and  AgN03  in 
solution. 


CHAPTER  XIX 


THERMOCHEMISTRY 

1.  Heat  of  Reaction. — It  may  be  stated  as  a general  rule  that 
all  physical  and  chemical  reactions  are  accompanied  by  heat 
effects.  The  heat  of  reaction  is  defined  to  be  the  number  of 
calories  of  heat  evolved  when  the  reaction  takes  place,  at  constant 
volume  in  the  direction  indicated,  and  between  the  amounts  of 
the  reacting  substances  indicated  by  the  stoichiometrical  equa- 
tion of  the  reaction.  Thus  the  equation, 

2H2+02  = 2HA)  + 117,400  (1) 

signifies  that  when  4 grams  of  hydrogen  gas  unite  in  a closed 
vessel  with  32  grams  of  oxygen  gas  to  form  36  grams  of  water 
vapor,  117,400  cal.  of  heat  are  evolved.  The  thermochemical 
equation  of  this  reaction  might,  if  desired,  be  written, 

H+JO  = | H^O  + 29,350  (2) 

which  is  the  thermochemical  equation  for  the  union  of  1 gram 
of  gaseous  hydrogen  with  8 grams  of  gaseous  oxygen  to  form 
9 grams  of  water  vapor.  A horizontal  line  above  the  formula 
of  a substance  in  a thermochemical  equation  indicates  that  the 
substance  is  understood  to  be  in  the  gaseous  state.  Similarly 
the  absence  of  any  line  indicates  the  liquid  state,  a line  below  the 
formula  indicates  the  crystalline  state,  and  a suffix,  aq  (thus 
HClag),  indicates  that  the  substance  is  in  solution  in  such  a large 
volume  of  water  that  the  addition  of  more  water  would  not  pro- 
duce any  appreciable  heat  effect,  that  is,  the  solution  is  under- 
stood to  be  so  dilute  that  its  heat  of  dilution  is  negligibly  small. 

Problem  1. — State  in  words  the  significance  of  the  following  thermo- 
chemical equations.  The  values  given  are  all  for  room  temperature. 

C -f-  02  = C02-f-  94,310 
C0  + 0=G02+68,000 

234 


(3) 

(4) 


Sec.  2] 


TIIERM0CHEM1STR  Y 


235 


CaO  + 2HCla(7  =CaCl2  ag-|-H20 +46,200  (5) 

Zn+H2S04a#  =ZnS04a<?+H2+37,700  (6) 

Ca+H20  =CaO  + H2+81.QOO  (7) 

Ca+2H20  =Ca(OH)2+H2+ 143,700  (8) 

Ca+2H20+ag  = Ca(OH)2ag+H2+94,100  (9) 

AgN03«2+KClag  = AgCl+KN03a?+ 15,800  (10) 

KCl-j-qq  = KClaq  — 4400  (11) 

HoO  = H20  - 1450  (at  18°)  (12) 

H20  = H»0  — 9990  (at  18°)  (13) 

Ci2H2>0n  + 1202  = 12C02+11H20  + 1,243,000  (14) 

C6H,+7*02  =6CO2+3HlO  + 754,300  (15) 

2.  Hess’s  Law. — 


Problem  2. — By  means  of  the  First  Law  of  -thermodynamics  (X,  4)  prove 
(1)  that  the  heat  of  a given  reaction  is  independent  of  whether  the  reaction 
actually  takes  place  as  written  or  whether  it  occurs  in  stages  and  (2)  if  it 
occurs  in  stages,  that  the  heat  of  the  reaction  is  independent  of  what  the 
stages  are  and  in  what  order  thejr  may  occur. 

These  facts  were  discovered  by  Hess  in  1840  before  the  First 
Law  of  thermodynamics  had  been  established  as  a general 
principle.  They  may  be  illustrated  by  the  process  of  forming  a 
dilute  solution  of  NH4C1  from  NH3  and  HC1.  This  process  may 
take  place  in  two  different  ways,  thus: 


and 

equals 

and 

and 

equals 


(1)  NH3+  HC1  = NH4Cl-f 42,100 

NH4Cl+a<7  = NH4Cla(7  — 3900 

NlL+HCl+ag  = NH4Clag+38,200 
(2)  NH3+a</  = NH3ag+8400 

HCl+ag  = HClag+ 17,300 


HCla2+NH3a<7  = NH4Clag+12,300 
NH3+ HC1  -\-aq  = NH4Clag+ 38,000 


(16) 

(17) 

(18) 

(19) 

(20) 
(21) 
(18) 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIX 


236 

It  is  evident  from  these  reactions  that  the  heat  of  the  whole 
reaction, 

NH3+HCl4-ag  = NH4Clag  (22) 

is,  within  the  experimental  error  (200  cal.),  the  same  in  both 

cases. 

Thermochemical  equations  can  be  added  and  subtracted,  or 
multiplied  and  divided  just  the  same  as  algebraic  equations. 
Hess’s  law  is  of  great  practical  importance  in  computing  heats 
of  reaction  for  cases  where  they  cannot  be  directly  measured. 

Problem  3. — From  the  thermochemical  equations  given  in  this  chapter 


calculate  the  heats  of  the  following  reactions: 

C+0  = CO+?  (23) 

Ca(OH)2+ttg  = Ca(OH)2ag+ ? (24) 

Ca(OH)2ag+2HClag  = CaC^ct^  -f-  2H20  T ? (25) 

772  + 0 = H20+?  (26) 


3.  Heat  of  Formation. — The  heat  of  formation  of  a compound 
is  the  heat  of  the  reaction  by  which  the  compound  is  formed  out 
of  its  elements.  Thus  the  heat  of  formation  of  cane  sugar, 
Ci2H220h,  out  of  solid  carbon  and  gaseous  oxygen  and  hydrogen 
would  be  the  heat  of  the  reaction 


12C  + 22H  + 110  = C12H220h  + ? (27) 

Problem  4. — Compute  the  heat  of  this  reaction  from  the  reactions  given 
above.  Compute  also  the  heat  of  formation  of  liquid  benzene,  CeH6,  from 
its  elements.  From  these  two  examples  formulate  a general  rule  for  calcu- 
lating the  heat  of  formation  of  a compound  of  C,  H and  O,  from  its  heat  of 
combustion.  Heats  of  combustion  are  comparatively  easy  to  measure  and 
through  them  it  is  possible  to  calculate  the  heats  of  many  other  reactions 
which  cannot  be  directly  measured. 

Problem  5. — From  the  heats  of  combustion  of  ethyl  alcohol  and  acetic  acid, 
respectively, 

C2H5OH  + 6O  = 2C02+3H20  + 325,700  (28) 

CH3COOH  +40  = 2C02+2H20  + 209,400  (29) 

calculate  the  heat  of  the  reaction 


C2H50H  + 02  = CH3C00H  + H20+? 


Sec.  5] 


THERMOCHEMISTRY 


237 


4.  Heats  of  Precipitation  and  of  Neutralization. 

Problem  6. — The  following  thermochemical  equations  are  true : (Ac  = the 


acetate  radical,  CH3COO). 

HClag + AgN  0 3aq  = AgCl + HNO  3a<?  +15, 800  (30) 

KC\aq+ AgN03ag  = AgCl+KN03ag+ 15,800  (31) 

NaClag+i  Ag2S04ag  = AgCl  + iNa2S04ag+ 15,800  (32) 

|BaCl2ag+ AgC103ag  = AgCl+ JBa(C103)2^+ 15,800  (33) 

CsClag+ AgAcaq  = AgCl-fCsAca<y+ 15,800  (34) 


Explain  in  terms  of  the  Ionic  Theory  why  the  heat  of  reaction  is  the  same  for 
all  of  the  above  reactions. 

Problem  7. — The  heat  of  neutralization  of  HC1  by  NaOH  in  dilute 
solution  is  expressed  by  the  thermochemical  equation 

HClag+NaOHa?  = NaClag+H2O  + 13,700  (35) 

By  means  of  the  Ionic  Theory  predict  the  heat  of  the.  following  reaction, 

LiOHatf + HN03a5  = LiN03ag + H20  + ? 

Problem  8. — -When  10  liters  of  0.1  N HC1  are  mixed  with  10  liters  of  0.1  N 
NaAc  (both  at  18°),  the  temperature  of  the  mixture  rises  0.0150°.  On  the 
assumption  that  the  specific  heat  capacity  (X,  2)  of  the  solution  is  equal  to 
unity,  calculate  the  heat  of  the  reacton.  What  is  the  heat  of  neutralization 
of  NaOH  by  HAc  in  0.1  A solution? 

(Suggestion:  Write  all  reactions  in  the  ionic  form  as  regards  the  strong 
electrolytes.  (Cf.  XVII,  3.)) 

5.  Heat  Effects  at  Constant  Volume  and  at  Constant  Pressure. 

— When  a reaction  takes  place  at  constant  volume,  the  heat 
v effect  ( Hv , cal.  evolved)  which  accompanies  it  is  by  definition 
understood  to  be  the  heat-of-the-reaction  (XIX,  1).  When  a re- 
action  occurs  without  change  of  volume  there  is  evidently  no- 
work  performed  against  the  external  pressure,  and  hence  by  the 
First  Law  of  thermodynamics  (X,  3) 

HV=-AU  (36) 

If,  however,  the  reaction  occurs  at  constant  pressure  P,  and 
is  accompanied  by  an  increase  of  volume  Av,  then  according  to  the 
S First  Law,  the  heat  effect  (Hp,  cal.  evolved)  at  constant  pressure 

differs  from  that  at  constant  volume  simply  by  the  work  PAv, 


238 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XIX 


performed  by  the  system  in  the  second  case  (X,  3 and  4,  equa- 
tion (3)),  that  is, 

Hv  — Hp  = PAv  (37) 

In  the  case  of  liquid  or  crystalline  systems,  is  always  small  and 
hence  in  most  cases  PAv  is  negligible  and  for  practical  purposes 
we  may  assume 

Hv  = Hp  (38) 

but  when  gases  are  involved  in  the  reaction,  Av  is  frequently 
large  and  the  work  term  must  be  taken  into  account. 

Problem  9. — If  A,  B,  M and  N are  perfect  gases,  show  that  for  the  reac- 
tion, aA+6B=mM-fnN  occurring  at  constant  pressure  P,  and  constant 
temperature  T,  the  work  performed  against  the  external  pressure  will  be 

PAv  = (m+n  — a — b)RT  (39) 

and  hence 

Hv  — Hp=  (ra+n  — a — b)RT  (40) 

Problem  10. — Calculate  in  calories  the  difference  between  Hv  and  Hv 
for  the  following  reactions:  (1)  2H2  + 02  = 2H20  at  20°;  (2)C  +02  = C02at 
2000°;  (3)  H20  = H20  at  5°;  (4)  H2Q  = H20  at  0°  and  1 atm.  The  density  of 
ice  is  0.9  gram  per  cubic  centimeter. 


CHAPTER  XX 


HEAT  CAPACITY 


CHAPTER  XXI 


CHEMICAL  KINETICS 


Homogeneous  Systems 

1.  Rate  of  Reaction. — The  rate  at  which  a chemical  reaction 
proceeds  in  a homogeneous  (I,  8)  system  is  defined  as  the  de- 
crease in  the  equivalent  concentration  of  the  reacting  molecular 
species  in  the  time,  dt,  divided  by  that  time,  or  mathematically 

dC 


rate  of  reaction  = 


d£ 


(i) 


Experiment  shows  that  in  gaseous  systems  and  in  dilute  so- 
lutions in  which  the  thermodynamic  environment  is  kept  con- 
stant the  rate  of  any  chemical  reaction  is  proportional  to  the 
product  of  the  concentrations  of  the  reacting  molecular  species, 
each  concentration  being  raised  to  a power  equal  to  the 
number  of  molecules  of  the  corresponding  molecular  species 
which  enter  into  the  reaction,  as  shown  by  the  chemical 
equation  which  expresses  the  reaction  as  it  actually  takes  place. 
Thus  if  the  reaction  whose  rate  is  being  measured  is  expressed  by 
the  equation 

aA+6B+  ....  ZjmM+nN+  ...  (2) 

then  the  law  just  stated  would  be  expressed  by  the  equation 


d[A]  d[B]  J.r.larp.,6 
- di=-  dt  =fc[A1  [B]  •• 


(3) 


where  the  brackets:  [A],  [B],  etc.,  signify  the  concentrations 
at  the  time,  t,  of  the  molecular  species,  A,  B,  etc.,  expressed  in 
equivalents  per  liter.  This  law  is  known  as  Guldberg  and  Waage’s 
law  of  chemical  mass  action  as  applied  to  reaction  rate. 

If  the  above  reaction  does  not  go  to  completion  in  the  direc- 
tion indicated  by  the  arrow,  the  rate  of  the  reverse  reaction 
must  also  be  considered.  This  would  evidently  be 

d[M]  cI[N] 

d£  d£ 

239 


= k'  [M]W[N]' 


(4) 


240 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXI 


and  the  resultant  rate  in  the  direction  indicated  by  the  arrow  would 
be  the  difference  between  the  actual  rates  of  the  two  opposing 
reactions,  that  is, 

_d^A]  = _d[A]+d[MJ  = fc[Ar[B]6  _V[Mr[N]»_  (5) 

In  what  follows  we  shall  confine  the  consideration  to  the  rates  of 
reactions  which  run  practically  to  completion  in  one  direction, 
that  is,  reactions  in  which  k'  is  negligibly  small. 

2.  First  Order  Reactions. — Reactions  involving  only  one  re- 
acting molecular  species  or  in  which  the  concentration  of  only 
one  reacting  substance  changes  are  called  first  order  reactions. 
The  change  of  dibrom-succinic  acid  into  brom-maleic  acid,  which 
occurs  when  it  is  boiled  with  water,  is  an  example  of  a reaction 
of  this  type. 

CHBrCOOH  CHCOOH 
| =|  + HBr 

CHBrCOOH  CBrCOOH 


Another  example  is  the  hydrolysis  of  cane  sugar  in  dilute  aqueous 
solution  in  the  presence  of  an  acid.  The  reaction  is 


C 12H22O11  “ H2O 


C6H12O6  CeH^Ofi 

glucose  ' fructose 


Problem  1. — Two  reacting  substances,  water  and  sugar,  evidently  are 
involved  in  this  reaction.  How  can  it  be  a first  order  reaction  according  to 
the  above  definition? 


For  a first  order  reaction  the  general  law  of  reaction  rate, 
equation  (3),  evidently  reduces  to  the  form 

_[JA1  = MA]  (6) 

or  if  we  call  A the  equivalent  concentration  of  the  reacting  sub- 
stance when  the  measurements  of  the  reaction  rate  are  begun 
(t.e.,  when  t = 0),  and  x the  number  of  equivalents  (per  liter) 
which  have  disappeared  as  a result  of  the  reaction,  at  the  end 
of  t units  of  time,  then  at  the  time  t we  have  [A ]=A—x  and 
— d[A]  = d^r,  and  equation  (6)  may  be  written  in  the  form 


(7) 


Sec.  3] 


CHEMICAL  KINETICS 


241 


Problem  2. — Show  that  the  integral  of  this  equation  is 

J logical- = 0.4343  k (8) 

Problem  3. — In  a solution  at  48°  containing  0.3  mole  of  cane  sugar  in  a 
liter  of  0.1  normal  HC1,  it  is  found  (by  means  of  a polarimeter)  that  32  per 
cent,  of  the  sugar  is  hydrolyzed  in  20  minutes,  (a)  Calculate  the  proportion- 
ality constant,  k (which  is  known  as  the  specific  reaction  rate),  (b)  Calculate 
the  rates  of  the  reaction  at  its  start  and  at  the  end  of  30  minutes,  (c)  What 
percent,  of  the  sugar  will  be  hydrolyzed  at  the  end  of  60  minutes?  (d) 
What  per  cent,  of  the  sugar  would  be  hydrolyzed  at  the  end  of  30  minutes 
if  the  0.3  mole  had  been  initially  dissolved  in  10  liters  of  the  0.1  N HC1  instead 
of  in  1 liter?  The  HC1  acts  merely  as  a catalyst  (XX,  12).  It  is  not  used  up 
during  the  reaction. 


3.  Second  Order  Reactions. — When  two  reacting  molecular 
species  or  two  molecules  of  the  same  species  simultaneously 
disappear  as  a result  of  the  reaction,  under  conditions  such 
that  the  concentrations  of  both  decrease,  then  the  reaction  is 
classed  as  a second  order  reaction. 

Examples: 

(1)  The  saponification  of  an  ester  by  an  alkali, 

CH3COOCH3  + NaOH  = CH3COONa+CH3OH 

(2)  The  polymerization  of  NO2  in  the  gaseous  state 

2N02  = N204 


Problem  4. — If  A is  the  initial  concentration  of  one  of  the  reacting  sub- 
stances and  B that  of  the  other,  and  x is  the  number  of  equivalents  per 
liter  of  each  which  have  been  changed  over  by  the  reaction  after  t units  of 
time,  show  that  the  equation  for  the  rate  of  the  reaction  may  be  written, 

^ = k{A-x){B-x)  (9) 

The  general  integral  of  this  equation  is 

(A-B)tlo^4jB^r0AUSk  <l0> 


Problem  5. — If  A and  B are  equal  or  if  the  two  reacting  molecules  are  of 
the  same  kind  show  that  the  integral  is 


1 X 

t (A—x)A 


= k 


(11) 


Problem  6. — In  the  saponification  of  methyl  acetate  (CH3COOCH3)  by 
caustic  soda,  20  per  cent,  of  the  ester  will  be  saponified  in  10  minutes  when  the 


242 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXI 


initial  concentrations  are  both  0.01  molal.  (a)  How  long  will  it  take  to 
saponify  99  per  cent.?  (b)  What  will  be  the  concentration  ot  methyl 
alcohol  at  the  end  of  half  an  hour?  If  the  initial  concentration  of  the 
ester  is  0.015  mole  and  that  of  the  NaOH  0.03  mole  per  liter,  (c)  what  per 
cent,  of  the  ester  will  be  saponified  in  10  minutes;  (d)  how  long  will  it  take  to 
saponify  99  per  cent,  and  (e)  what  will  be  the  concentration  of  methyl  alcohol 
at  the  end  of  half  an  hour? 

4.  Third  Order  Reactions. — This  order  includes  all  reac- 
tions in  which  three  reacting  molecules  are  directly  involved 
with  concentration  decrease. 

Example: 

2CH3C00Ag+HC00Na  = 2Ag+C02+CH3C00H+ 

CH3COONa 

The  three  molecules  here  are  two  molecules  of  silver  acetate  and 
one  of  sodium  formate.  The  equation  for  the  rate  of  this  re- 
action would  evidently  be 

^ = k(A-x)KB-x)  (12) 

5.  Reactions  of  Higher  Orders. — Reactions  of  higher  orders 
than  the  third  are  of  very  rare  occurrence.  For  example,  the 
reaction  which  is  written 

2PH3+402  = P205  + 3H20 

represents  the  combustion  of  the  gas  PH3.  This  would  appar- 
ently be  a reaction  of  the  sixth  order  since  six  reacting  molecules 
are  involved.  Experiment  shows,  however,  that  the  rate  of 
this  reaction  actually  corresponds  to  the  equation  of  a second 
order  reaction.  The  interpretation  placed  upon  this  fact  is 
that  the  above  reaction  actually  occurs  in  stages.  The  first 
stage  is  the  slow  reaction,  PH3  + 02  = HP02+H2.  This  is 
evidently  a second  order  reaction.  The  subsequent  reactions 
by  which  H20  and  P205  are  produced  are  very  rapid  in  compari- 
son with  the  first  stage  so  that  the  rate  which  is  actually  meas- 
ured experimentally  is  the  rate  of  the  first  stage  of  the  reac- 
tion, which  accounts  for  the  fact  that  the  reaction  behaves  as  a 
second  order  one.  The  rate  of  a reaction  which  occurs  in  stages 
can  obviously  never  be  greater  than  the  rate  of  the  slowest  stage 
and  this  is  the  reason  why  reactions  of  higher  orders  are  so  rarely 


Sec.  7] 


CHEMICAL  KINETICS 


243 


met  with.  The  slowest  stage  of  the  reaction  determines  its 
order  and  the  slow  stages  are  practically  always  either  first, 
second  or  third  order  reactions. 

6.  Saponification  and  the  Ionic  Theory.— 

Problem  7. — (a)  In  dilute  solution  the  rate  of  saponification  of  ethyl 
acetate  by  Ba(OH)2,  Ca(OH)2,  or  Sr(OH)2  is  found  to  obey  the  equation  of 
a second  order  reaction  and  to  have  a specific  reaction  rate  only  slightly 
smaller  than  in  the  case  of  saponification  with  NaOH.  Stoichiometrically 
the  reaction  would  be  written 

2CH3COOC2H5+Ba(OH)2  = (CH3COO)2Ba+2C2H5OH 

and  would  appear  to  be  a third  order  reaction.  Show  how  according  to  the 
Ionic  Theory  we  should  expect  the  reaction  to  be  a second  order  one  and 
to  have  nearly  the  same  specific  reaction  rate  as  in  the  case  of  NaOH. 
(b)  The  specific  reaction  rates  for  saponification  with  KOH,  LiOH  and  CsOH 
are  the  same  as  for  NaOH  but  that  for  NH4OH  is  very  much  smaller  (ini- 
tially only  about  as  large,  in  fact,  for  a 0.1  normal  solution).  All  the 
reactions  are  second  order  ones.  How  are  these  facts  interpreted  by  the 
Ionic  Theory?  (Suggestion:  Write  all  the  reactions  in  the  ionic  form 
and  make  use  of  the  facts  stated  in  XVII,  3.  Esters  are  non-electrolytes.) 

Problem  8. — At  25°  the  initial  rate  of  saponification  of  ethyl  acetate  by 
0.01  normal  NaOH  is  9.0  times  as  great  as  that  by  0.1  normal  KCN.  From 
the  conclusions  reached  in  problem  7,  together  with  the  data  given  in  XVII, 
3 and  Table  XXIII,  what  would  you  conclude  as  to  the  concentration  of 
hydroxyl  ion  in  the  KCN  solution?  The  source  of  this  hydroxyl  ion  will  be 
explained  later.  (XXIII,  3.) 

7.  Reaction  Rate  and  Thermodynamic  Environment.  The 
Neutral  Salt  Effect. — Reaction  rates  in  solution  are  usually 
appreciably  influenced  by  the  presence  of  neutral  salts  which 
apparently  have  nothing  directly  to  do  with  the  reaction  itself. 
In  many  cases  this  “ neutral  salt  effect,”  as  it  is  called,  may  be 
interpreted  as  due,  at  least  partially,  to  the  influence  of  the  ions 
of  the  salt  upon  the  thermodynamic  environment  (XIII,  3)  pre- 
vailing within  the  solution,  for  the  rate  of  any  given  reaction  is 
very  powerfully  influenced  by  the  nature  of  the  medium  in  which 
it  takes  place.  In  employing  reaction  rates  for  determining  the 
concentration  of  some  molecular  species  in  a given  solution  it  is, 
therefore,  essential  that  the  experiments  be  conducted  as  far  as 
possible  under  comparable  conditions  as  regards  thermodynamic 
environment.  Thus  in  problem  8,  the  NaOH  solution  employed 
in  the  experiment  should  have  been  one  which  had  an  OH~-ion 


244 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXI 


concentration  as  close  as  possible  to  that  prevailing  in  the  KCN 
solution  and  it  should  also  have  contained  sufficient  KC1  to 
make  its  total  concentration  of  uni-univalent  electrolyte,  0.1 
equivalent  per  liter.  The  thermodynamic  environment  within 
the  NaOH  solution  would  then  probably  have  been  as  close 
to  that  prevailing  within  the  KCN  solution  as  it  is  possible  to 
make  it.  Only  when  the  thermodynamic  environments  are 
identical  in  two  solutions  are  we  justified  in  assuming  that  a 
given  reaction  has  the  same  specific  reaction  rate  in  both  solu- 
tions. This  should  be  borne  constantly  in  mind  in  employing 
the  laws  of  chemical  kinetics  in  the  study  of  solutions. 

8.  Temperature  and  Reaction  Rate. — It  has  been  found  in 
many  cases  that  equal  increments  of  temperature  produce 
about  the  same  multiplication  of  reaction  rate.  For  each 
rise  of  10°  the  rate  of  the  reaction  is  multiplied  2-4  fold,  the 
exact  value  of  the  multiplying  factor  varying  with  the  nature 
of  the  reaction. 

Problem  9. — Assuming  2.5  as  an  average  value  for  this  multiplying  factor 
calculate  how  many  fold  the  reaction  rate  would  be  increased  by  a rise  of 
100°. 

A more  general  expression  for  the  effect  of  temperature  upon 
the  specific  reaction  rate  is  that  formulated  by  van’t  Hoff  as 
follows: 

d log  k A . 

d T ~ T 2 { } 

where  A is  a characteristic  constant. 

Catalysis 

9.  The  Phenomenon  of  Catalysis. — The  rate  of  chemical  reac- 
tions in  many  instances  can  be  greatly  increased  by  the  presence 
in  the  reaction  mixture  of  substances,  called  catalytic  agents 
or  catalysts,  which  are  not  themselves  consumed  by  the  re- 
action. This  phenomenon  is  known  as  catalysis.  There  are 
many  kinds  and  varieties  of  catalysts  and  the  mechanism  of 
their  action  is  so  different  in  different  cases  and  so  little  under- 
stood that  few  if  any  general  principles  can  be  laid  down  concern- 
ing the  phenomenon.  It  is  of  immense  importance,  however, 
in  scientific  and  industrial  processes  as  well  as  in  many  processes 


Sec.  13] 


CHEMICAL  KINETICS 


245 


which  occur  in  nature,  and  a brief  statement  regarding  what  is 
known  concerning  the  action  of  some  classes  of  catalysts  will  be 
included  here. 

10.  Contact  Agents. — Many  reactions  both  in  the  gaseous 
state  and  in  solution  are  greatly  accelerated  near  the  surface  of 
certain  solids  such  as  platinum  black,  ferric  oxide  and  other 
metallic  oxides.  Such  substances  in  a finely  divided  state 
adsorb  (that  is,  concentrate  upon  their  surface)  the  reacting  sub- 
stances from  the  gas  or  solution  and  the  reaction  in  the  adsorbed 
layer  then  proceeds  much  more  rapidly  than  in  the  body  of  the 
gas  or  solution  where  the  concentrations  are  much  lower.  This 
is  supposed  to  be  the  mechanism  of  the  catalytic  effect  of  some 
of  the  contact  agents.  The  action  appears  in  many  cases, 
however,  to  be  a very  specific  one.  The  contact  process  of 
sulphuric  acid  manufacture  is  an  example  of  a technical  appli- 
cation of  a contact  agent. 

11.  Carriers. — In  many  cases  the  catalyst  actually  reacts 
chemically  with  one  of  the  reacting  substances  to  form  an  inter- 
mediate compound  which  in  turn  reacts  in  such  a way  as  to  re- 
generate the  catalyst  and  produce  the  final  reaction  products. 
Such  catalysts  are  known  as  carriers.  Thus  nitric  oxide,  NO, 
acts  as  a carrier  in  the  familiar  chamber  process  for  the  manu- 
facture of  sulphuric  acid. 

02+2N0  = 2N02 

and  then 

so2+no2+ h2o =h2so4+no 

12.  Ions  as  Catalysts. — Hydrogen  ion  and  hydroxyl  ion  act  as 
catalysts  for  many  reactions  whch  occur  in  aqueous  solution. 
The  velocity  of  any  reaction  catalyzed  by  either  of  these  two 
ions  is,  in  dilute  solution,  directly  proportional  to  the  con- 
centration of  the  ions  in  question  as  long  as  the  thermodynamic 
environment  remains  constant.  The  inversion  of  cane  sugar 
by  an  acid,  the  hydrolysis  of  an  ester  by  an  acid  and  the  hydra- 
tion of  milk  sugar  by  an  alkali  are  examples  of  such  reactions. 
They  are  frequently  employed  for  the  purpose  of  determining 
the  concentration  of  the  catalyzing  ion  in  a given  solution. 

13.  Enzymes. — Many  animal  and  vegetable  organisms  secrete 
certain  complex  colloidal  compounds,  known  as  enzymes,  which 


246 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXI 


possess  highly  specific  but  very  powerful  catalytic  powers. 
Thus  the  yeast  plant  secretes  the  enzyme,  invertase,  which  is 
able  to  convert  cane  sugar  into  glucose  and  fructose,  while 
another  yeast  enzyme  known  as  xymase  will  convert  glucose 
(but  not  the  closely  related  compound,  fructose)  into  alcohol  and 
carbon  dioxide.  On  the  other  hand  glucose  will  be  converted 
into  lactic  acid  by  an  enzyme  produced  by  the  lactic  acid  bacillus. 
Enzymes  are  responsible  for  many  of  the  processes  of  physiology 
(as  pepsin,  lypase,  and  trypsin  in  digestion)  and  for  the  processes 
of  decay  of  dead  animal  and  vegetable  matter.  The  velocity 
of  reactions  catalyzed  by  enzymes  increases  with  the  tempera- 
ture up  to  a point  usually  between  30  and  40°  after  which 
it  decreases  owing  to  the  destruction  of  the  enzyme  by  heat. 
Enzymes  are  employed  in  a number  of  industrial  processes  such 
as  brewing  and  butter  and  cheese  manufacture. 

14.  Water  as  a Catalyst. — The  presence  of  water  seems  to  be 
indespensible  for  the  occurrence  of  many  reactions.  Thus  HC1 
and  NH3  when  brought  together  in  a perfectly  dry  condition  will 
not  react  with  each  other.  The  presence  of  the  merest  trace  of 
water  vapor,  however,  suffices  to  start  the  reaction.  The 
mechanism  of  this  action  is  not  understood  but  it  seems  to  re- 
semble in  some  respects  the  effect  of  a percussion  cap  in  setting 
off  an  explosive.  Once  started  the  reaction  is  able  to  proceed 
of  itself  without  further  assistance. 

15.  Autocatalysis. — Some  reactions  are  catalyzed  by  the 
presence  of  one  or  more  of  their  own  reaction  products  and  are, 
therefore,  said  to  be  autocatalyzed.  The  reaction  between  oxalic 
acid  and  a permanganate  in  solution  is  catalyzed  in  this  manner. 
The  decomposition  of  Ag20  into  Ag  and  02  at  high  temperatures 
is  catalyzed  by  the  metallic  silver  and  the  rate  of  the  reaction 
increases  with  the  amount  of  silver.  The  action  in  this  instance, 
however,  is  probably  a contact  action,  the  reaction  20  = 02 
being  catalyzed  at  the  surface  of  the  finely  divided  silver. 

16.  Radiant  Energy. — Many  reactions  which  proceed  very 
slowly  or  inappreciably  in  the  dark  are  greatly  accellerated  when 
illuminated  by  radiant  energy  of  the  proper  wave  length.  The 
union  of  H2  and  012  gases  and  the  changes  which  take  place  upon 
a photographic  plate  are  examples  of  photochemical  reactions. 
Not  only  is  the  reaction  rate  influenced  by  the  light,  but  the  nature 


Sec.  17] 


CHEMICAL  KINETICS 


247 


of  the  reaction  products  and  the  yield  obtained  are  also  affected. 
The  subject  of  photochemistry  is  being  extensively  investigated 
at  the  present  time.  Other  forms  of  energy  such  as  the  silent 
electric  discharge,  X-rays,  cathode  rays,  etc.,  also  exert  a power- 
ful influence  upon  the  course  and  rate  of  chemical  reactions  in 
many  cases.  These  effects  while  introduced  at  this  point,  are 
not  usually  classed  as  catalytic,  however. 

Heterogeneous  Systems 

17.  Solid  Substances. — When  a solid  substance  reacts  with  a 
substance  in  solution  the  rate  of  the  reaction  is  proportional  to 
the  surface  of  the  solid  exposed  to  the  solution.  When  a solid 

dissolves  in  its  own  solution  the  rate  at  which  it  dissolves,  .7 

at 

under  uniform  conditions  of  temperature  and  of  stirring  is  pro- 
portional to  the  surface,  A,  exposed  and  to  the  difference  between 
its  solubility,  S,  (XIV,  12),  and  its  concentration  C,  in  the  solu- 
tion at  the  time  t,  or  mathematically 

a±=kA(S-C)  (14) 

Problem  10. — (a)  What  are  the  relative  rates  at  which  a substance  will 
dissolve  in  its  own  solution  (1)  when  the  solution  is  50  per  cent,  saturated 
and  (2)  when  it  is  98  per  cent,  saturated?  (b)  If  it  takes  2 minutes  for 
the  solution  to  become  50  per  cent,  saturated,  how  long  should  a solubility 
experiment  last,  if  one  wishes  to  determine  the  solubility  with  an  accuracy 
of  0.01  per  cent.?  Assume  constant  surface  and  constant  conditions  of 
stirring. 


CHAPTER  XXII 


CHEMICAL  EQUILIBRIUM 

A.  Homogeneous  Systems  at  Constant  Temperature 
1.  The  Nature  of  Chemical  Equilibrium. — Suppose  that  we 
bring  together  in  any  homogeneous  gas  or  liquid  system  two 
substances  A and  B which  enter  into  chemical  reaction  with 
each  other  to  form  the  substances  M and  N as  expressed  by  the 
equation 

aA+EB  = mM+wN  (1) 

As  this  reaction  proceeds,  the  concentrations  of  A and  B will  be 
observed  to  decrease  and  those  of  M and  N to  increase,  at  first 
rapidly  and  then  more  slowly  until  finally  a steady  condition 
will  be  reached  in  which  the  concentrations  of  the  molecular 
species  A,  B,  M and  N no  longer  change,  that  is,  the  reaction 
has  apparently  stopped.  In  the  same  way  if  we  were  to  start 
with  the  two  substances  M and  N at  the  same  initial  equivalent 
concentrations  as  were  used  for  A and  B in  the  first  experiment, 
we  would  find  that  reaction  (1)  would  proceed  from  right  to  left 
until  eventually  a steady  condition  of  no  concentration  change 
would  again  be  reached.  The  concentrations  of  the  four  molecu- 
lar species  when  this  steady  state  is  attained  are  found  to  be 
independent  of  the  direction  from  which  this  condition  is  ap- 
proached. That  is,  whether  we  started  with  a moles  of  A and 
h moles  of  B or  with  m moles  of  M and  n moles  of  N,  the  con- 
centrations of  all  four  species  when  the  reaction  had  apparently 
ceased  would  have  the  same  respective  values  regardless  of  the 
direction  in  which  the  reaction  took  place.  When  a steady 
condition  of  this  character  is  reached  in  any  chemical  reaction, 
the  molecular  species  involved  are  said  to  be  in  chemical  equilib- 
rium with  one  another.  This  equilibrium  is  not  a static  one, 
however,  but  is  instead  a dynamic  one.  That  is,  the  chemical 
reaction  does  not  actually  cease  but  rather  it  continues  to  take 
place  in  both  directions  but  at  the  same  rate  in  each  direction. 

248. 


Sec.  2] 


CHEMICAL  EQUILIBRIUM 


249 


The  reacting  substances  are  thus  reformed  by  the  reverse  re- 
action just  as  fast  as  they  are  used  up  by  the  direct  reaction,  the 
net  concentration  change  being  zero. 

The  concentrations  of  the  reacting  molecular  species  when 
equilibrium  is  attained  will  be  called  their  equilibrium  concen- 
trations. In  any  chemical  equilibrium  in  a perfect  gas  or  a 
dilute  solution  at  a given  temperature  the  equilibrium  concen- 
trations, CA,  CB,  CM  and  C N.,  are  always  so  related  that  they 
fulfill  the  following  condition 


Cm  sin 
M*^N 


ci-d 


= const. 


(2) 


where  the  value  of  the  11  equilibrium  constant,”  as  it  is  called,  is, 
for  a given  thermodynamic  environment,  characteristic  of  the 
reaction  in  question.  This  relation  is  Guldberg  and  Waage’s 
law  of  chemical  mass  action  as  applied  to  chemical  equilibrium. 
In  section  3 we  shall  recognize  it  as  a special  case  of  a more 
general  law  based  upon  the  principles  of  thermodynamics. 

2.  The  Criterion  for  True  Chemical  Equilibrium. — The  fact 
that  in  any  instance  a given  chemical  reaction  has  apparently 
come  to  a stop  cannot  be  taken  as  proof  that  a true  chemical 
equilibrium  has  been  attained.  In  order  to  make  certain  the 
equilibrium  should  be  approached  from  both  directions.  If  the 
value  of  the  equilibrium  constant  computed  from  the  equilibrium 
concentrations  is  found  to  be  independent  of  the  direction  from 
which  the  equilibrium  is  approached,  then  the  equilibrium  is  a 
ture  chemical  equilibrium.  False  equilibria  are  sometimes  met 
with,  due  to  various  causes  such  as  destruction  or  absence  of  an 
essential  catalytic  agent  (XXI,  9),  extreme  slowness  of  some  stage 
of  the  reaction,  or  slow  diffusion  of  one  or  more  of  the  substances 
involved.  False  equilibria  occur  most  frequently  in  the  case  of 
reactions  in  heterogeneous  systems. 


Problem  1. — In  one  experiment  0.100  mole  each  of  H2  and  C02  are 
heated  together  at  686°  in  a closed  liter  vessel  until  the  reaction  has  appar- 
ently ceased.  The  vessel  is  then  found  to  contain  0.0422  mole  of  H20.  In 
a second  experiment  0.110  mole  of  CO  and  0.0902  mole  of  H20  are  heated 
in  the  same  way  and  when  the  reaction  has  apparently  ceased  0.0574  mole 
of  C02  is  found  to  have  been  formed.  Was  true  chemical  equilibrium 
attained  in  these  exponents? 


250 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


3.  The  Thermodynamic  Law  of  Chemical  Equilibrium  at 
Constant  Temperature. — (a)  Gaseous  Systems.  If  any  number 
of  molecular  species  A,  B,  . . . . , M,  N,  . . etc.,  are  in  chemical 
equilibrium  with  one  another  in  the  gaseous  state  as  expressed 
by  the  equation 

aA+6B+  . . mM+nN+ (3) 

the  following  purely  thermodynamic  (XII,  8)  relationship  can 
be  shown  to  connect  the  partial  pressures,  p,  and  molal  volumes, 
v,  of  the  molecular  species  concerned,  at  any  given  constant 
temperature  T. 

m^M(lpM+^NdpN+ • • . —avAdpA  — bvBdpB— . . .=0  (4) 
where  dp  for  each  substance  represents  the  complete  differential 

dp=  dP+  (^)  dx,  P being  the  total  pressure  on  the  sys- 

tem and  x the  mole  fraction  of  any  constituent  of  the  system 
whether  it  be  concerned  in  the  chemical  reaction  or  not.  In 
other  words,  if  the  chemical  equilibrium  be  displaced  in  one  direc- 
tion or  the  other  by  changing  either  the  pressure  or  the  composi- 
tion of  the  gas  or  both,  the  corresponding  changes  in  the  partial 
pressures  of  the  molecular  species  concerned  in  the  equilibrium 
must  occur  in  such  a way  as  to  fulfill  the  condition  represented  by 
equation  (4),  which  for  brevity  may  be  written 

'L±mvMdpwi  = 0 (5) 

where  the  + sign  indicates  that  the  terms  for  the  substances  on 
one  side  of  the  reaction  must  be  taken  with  opposite  signs  from 
those  for  the  substances  on  the  other  side  of  the  reaction. 

In  order  to  integrate  this  equation  it  is  first  necessary  to  know 
the  equation  of  state  for  each  molecular  species  involved.  If 
all  the  substances  obey  the  perfect  gas  law  or  in  geheral  if  p 
represents  thefugacity  (XIV,  1),  we  have  mvMdpM  = mRTd\og>ePM 
= RT  dlogepS  and  similarly  for  each  of  the  other  substances 
involved  in  the  reaction.  Equation  (5)  therefore  becomes 


RT  dloge 

which  on  integration  gives 


„jn  n 

Pm  V n 


Vm'Vn 


a b 

Pa'Pb 


= 0 


= const.  = K 


(6) 


(7) 


Sec.  3] 


CHEMICAL  EQUILIBRIUM 


251 


\ 


This  is  the  law  of  chemical  mass  action  (for  a gaseous  system) 
expressed  in  terms  of  partial  pressures  or  fugacities.  If  the  sub- 
stances are  all  perfect  gases,  we  may  also  write  Pm  = (CmRT)™ 

and  similarly  for  each  of  the  other  substances,  where  C (^  = ^ 


represents  the  equilibrium  concentration  in  moles  per  liter 
(XXII,  1)  in  each  instance.  Equation  (7)  may  therefore  be 
written 


Cm  pin 
M'^N  ■ 
pa  pb 


= KP(RT)An  = Kc 


(8) 


where 


An  = a-\-b-{-.  . . —m  — n—  . 


(9) 


and  we  thus  obtain  the  mass  action  law  (equation  2)  in  terms  of 
volume  concentrations.  The  relationship  between  the  equilib- 
rium constants,  Kv  and  Kc,  of  the  two  modes  of  expression  is 
shown  in  equation  (8)  from  which  one  constant  can  evidently 
be  calculated  if  the  other  is  known. 


Problem  2. — Show  that  for  perfect  gases  the  mass  action  law  may  also  be 
written  in  terms  of  the  mole  fractions  of  the  substances  concerned,  thus 


x^xnN 
x\  -Xb 


= KpPAn  = KX  (if  P is  constant) 


(10) 


(b)  Liquid  Systems — In  the  case  of  a.  chemical  equilibrium 
in  solution,  equation  (7)  where  p represents  fugacity,  must 
evidently  hold  for  the  vapor  above  the  solution  and  if  the 
thermodynamic  environment  in  the  solution  is  constant,  the 
fugacity  of  each  molecular  species  will  be  proportional  to  its 
mole  fraction,  x,  in  the  solution  (see  equation  1,  XIV).  Equa- 
‘ tion  (7)  may,  therefore,  be  written 

^M'^N 

a h = const.  =KX  (11) 

Xx'Xb 


for  the  solution.  This  is  the  law  governing  a chemical  equilib- 
rium in  any  solution  in  which  the  thermodynamic  environment 
is  constant. 


Problem  3. — Show  that  from  the  nature  of  the  definitions  (XI,  5)  of  x 
and  C for  any  substance  in  a solution  the  following  relationship  exists 


Cm  pn 

M’t'  N 


Cl-c ] 


— ^x(C,S  + CrA  + ^B+  • . • +Cm  + Cn+  • • - Y 


(12) 


252 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


where  An  and  Kx  are  defined  by  equations  (9)  and  (11)  respectively  and  Cs 
represents  the  total  concentration  (in  moles  per  liter)  of  all  molecular  species 
in  the  solution  which  are  not  directly  concerned  in  the  reaction  (solvent 
molecules,  for  example). 

Problem  4. — Under  what  mathematical  condition  and  also  under  what 
other  practical  condition  will  equation  (12)  reduce  to  the  form 


rvn 

Ca'Cb 


— const.  = Kc 


(13) 


In  writing  the  mass  action  law  in  terms  of  concentrations  it  is  frequently 
customary  to  employ  the  formula  of  the  substance  inclosed  in  brackets  as  a 
symbol  representing  its  volume  concentration.  Thus  in  this  nomenclature 
equations  (2)  and  (13)  would  be  written 

[M]m  [N]" 

[A]a  [B]6’  • • Kc 


4.  Determination  of  the  Equilibrium  Constant. — In  order  to 
determine  by  direct  measurement  the  numerical  value  of  the 
equilibrium  constant  for  a given  reaction  at  a given  temperature 
it  is  first  necessary  to  establish  equilibrium  between  the  different 
substances  taking  part  in  the  reaction  and  then  to  measure  the 
concentration  of  each  substance  in  this  equilibrium  mixture.  I11 
order  to  be  certain  that  true  equilibrium  has  been  attained  it 
should,  as  explained  above,  be  approached  from  both  directions 
and  in  many  cases  a catalyst  (XX,  9)  must  be  used  in  order  to 
hasten  the  attainment  of  equilibrium.  The  method  employed 
for  determining  the  concentration  of  any  substance  in  an  equilib- 
rium mixture  must  be  one  which  does  not  change  this  concen- 
tration during  the  process  of  determining  its  value,  for  such  a 
change  would  result  in  a displacement  of  the  equilibrium.  The 
customary  methods  of  chemical  analysis  cannot,  therefore,  be 
employed  in  many  cases  and  we  are  obliged  to  modify  these 
methods  or  to  resort  to  physical  methods  for  measuring  the 
equilibrium  concentrations  of  the  molecular  species  concerned. 

As  an  example  we  will  consider  the  problem  of  determining  the 
equilibrium  constant  for  the  gaseous  reaction 

2HI  H2TT2 

Equilibrium  in  this  reaction  may  be  reached  either  by  heating 
gaseous  HI  to  the  desired  temperature  or  by  heating  a mixture 
of  H2  and  I2  gases  to  the  same  temperature.  This  equilibrium 


Sec.  4J 


CHEMICAL  EQUILIBRIUM 


253 


is  attained  very  slowly  at  ordinary  temperatures  but  at  the 
temperature  of  440°  it  is  reached  in  a short  time.  If,  therefore, 
we  heat  a known  quantity  of  HI  (or  of  H2  and  I2)  at  440°  in  an 
equilibrium  vessel  of  known  volume  until  equilibrium  is  reached, 
we  may  determine  the  equilibrium  concentrations  by  drawing 
out  a sample  of  the  mixture  through  a cooled  capillary  tube  con- 
nected with  the  equilibrium  vessel  and  then  analyzing  this  sample 
at  our  leisure,  for  the  mixture  on  being  drawn  into  the  capillary 
is  cooled  so  quickly  that  the  rate  of  reaction  is  reduced  nearly 
to  zero  before  any  appreciable  shifting  of  the  equilibrium  can 
occur  and  the  composition  of  the  cold  sample  will  be  the  same  as 
that  of  the  equilibrium  mixture  at  the  higher  temperature.  In 
one  experiment  of  this  kind  Bodenstein  heated  20.55  moles  of 
H2  and  with  21.89  moles  of  I2  to  440°  until  equilibrium  was  at- 
tained. The  equilibrium  mixture  was  then  rapidly  cooled  and  on 
analysis  (by  absorbing  the  I2  and  HI  in  caustic  potash  and  meas- 
uring the  H2  remaining)  was  found  to  contain  2.06  moles  of 
H2,  13.4  moles  of  I>  and  37.0  moles  of  HI  in  the  volume,  v. 
Hence 


[H2]  [I2]  1 

mi 

m 

[HI]2  ' 

I"3' 

T 

0.020 


(15) 


Instead  of  cooling  such  an  equilibrium  mixture  and  then  ana- 
lyzing it,  it  is  frequently  possible  to  determine  its  composition 
at  equilibrium  by  a suitable  physical  method.  Thus  in  the 
present  case  HI  and  H2  are  both  colorless  gases  while  I2  is  strongly 
colored  and  by  comparing  the  depth  of  color  of  the  equilibrium 
mixture  with  that  of  a series  of  vessels  containing  iodine  vapor 
alone,  at  known  concentrations,  the  composition  of  the  equilib- 
rium mixture  can  be  determined. 

Problem  5. — A one  liter  quartz  tube  containing  0.1  mole  of  HI  gas  is 
heated  to  440°  until  equilibrium  is  established.  By  comparing  the  depth 
.of  the  violet  color  observed  on  looking  through  the  tube,  with  the  color  of 
standard  tubes  containing  pure  iodine  vapor,  the  color  of  the  equilibrium 
mixture  is  found  to  correspond  to  an  I2  concentration  of  0.0112  mole  per 
liter.  From  these  data  compute  the  equilibrium  constant  for  this  equilib- 
rium. (Cf.  XI,  problem  1.) 


254 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


Problem  6. — A liter  vessel  containing  0.14  mole  of  H2  and  20.6  grams  of 
iodine  is  heated  to  440°.  How  much  HI  will  be  formed?  What  will  be  the 
partial  pressure  of  each  substance  when  equilibrium  is  reached?  What  will 
be  the  total  pressure  of  the  mixture? 

Problem  7. — x grams  of  HI  are  heated  in  a 2-liter  vessel  at  440°  until 
equilibrium  is  reached,  the  total  pressure  being  0.5  atmosphere.  Calcu- 
late x. 


When  a gas  containing  more  than  one  atom  in  its  molecule 
is  heated  to  a sufficiently  high  temperature  it  breaks  up  or  dis- 
sociates partly  into  simpler  molecules  (XX,  4)  and  the  fractional 
extent  to  which  such  dissociation  occurs  at  any  given  temperature 
and  pressure  is  called  the  degree  of  dissociation,  a,  of  the  gas. 
This  degree  of  dissociation  and  hence  also  the  equilibrium  con- 
stant of  the  reaction  can  usually  be  calculated  by  determining 
the  density  of  the  partially  dissociated  gas  in  equilibrium  with 
its  dissociation  products  at  the  temperature  in  question  and 
comparing  this  observed  density,  D0 bs.,  with  the  density,  Dc aic., 
which  the  gas  would  have  at  the  same  temperature,  if  it  did  not 
dissociate.  This  latter  density  can  of  course  be  calculated  from 
the  molecular  weight  of  the  undissociated  gas  and  the  tempera- 
ture and  pressure. 

% 

Problem  8. — When  one  molecule  of  a gas  dissociates  it  produces  n mole- 
cules of  dissociation  products.  If  one  mole  of  the  gas  is  heated  to  such  a 
temperature  that  the  degree  of  dissociation  is  a , show  that  the  total  number 
of  moles  of  all  molecular  species  present  in  the  dissociation  mixture  is  1 + 
{n  — l)a.  Show  also  that  a can  be  calculated  from  the  equation 


a = 


D 


calc* 


D 


obs* 


(yI  l)Z)0bs* 


(16) 


Problem 

equation 


9. — N2O4  dissociates  in  the  gaseous  state  according  to  the 
N2O4  = 2N02 


At  49.7°  and  26.8  mm.  the  observed  density  of  the  gas  referred  to  air  is 
1.663.  Calculate  its  degree  of  dissociation. 

Problem  10. — Show  that  for  any  dissociation  of  the  type,  AB=A-fB 
or  A2  = 2A,  the  equilibrium  constant,  Kc,  for  the  dissociation  reaction  is 
expressed  by  the  relation 


(17) 


where  C is  the  volume  concentration  of  the  dissociating  substance  calculated 
on  the  assumption  that  it  does  not  dissociate. 


Sec.  5] 


CHEMICAL  EQUILIBRIUM 


255 


Problem  11. — From  the  data  in  the  preceding  problems  calculate  the 
value  of  Kc  for  the  dissociation  of  N204  at  49.7°.  Calculate  also  Kp,  and 
m Kx  for  one  atmosphere  pressure.  From  your  results  calculate  a for  P = 
1,  for  P = 0.5,  and  for  P = 0.1  atmosphere  respectively.  What  would  be 
the  partial  pressure  of  the  N02  molecules  for  P = 0.6  atmosphere.  Cal- 
culate the  actual  density  (referred  to  air)  of  the  gas,  for  P =93.75  mm. 
(The  observed  value  at  this  pressure  is  1.79.) 

Problem  12. — Phosphorous  pentachloride  dissociates  in  the  gaseous 
state  according  to  the  equation 
* PC15=PC13+C12 

When  1 gram  of  PCU  crystals  is  vaporized  at  274°  and  1 atmosphere  the 
density  of  the  vapor  referred  to  air  is  3.84  when  equilibrium  is  reached. 
Calculate  a,  Kc,  Kp  and  Kx.  Calculate  the  degree  of  dissociation  of  the 
PC15  after  equilibrium  is  reached  in  each  of  the  following  experiments:  (a) 
0.1  gram  of  Cl2  is  mixed  with  the  PCI5,  (1)  at  constant  volume  and  (2)  at 
constant  pressure;  (b)  the  same  experiments  with  0.1  gram  of  Ar;  (c)  the 
same  experiments  with  0.1  gram  of  PC13;  (d)  the  total  pressure  is  reduced  to 
0.5  atmosphere;  (e)  the  volume  is  increased  to  50  liters  by  reduction  of 
pressure. 

Problem  13. — At  290°  and  330°  respectively  the  sublimation  pressure 
(VI,  1)  for  NH4CI  crystals  is  185.3  and  610.6  mm.  respectively.  The  meas- 
ured values  for  the  density  of  the  saturated  vapor  at  the  same  two  tempera- 
tures are  0.00017  and  0.00053  grams  per  cubic  centimeter  respectively. 
Calculate  the  degree  of  dissociation  and  the  equilibrium  constant  for  each 
temperature. 

5.  Chemical  Equilibrium  in  Solutions  of  Constant  Thermo- 
dynamic Environment. — All  chemical  equilibria  in  solutions 
where  the  thermodynamic  environment  remains  constant  must 
obey  the  law  of  mass  action  as  expressed  by  equations  (11)  and 
(13)  since  for  such  solutions  this  law  a purely  thermodynamic 
deduction.  The  application  of  the  mass  action  law  to  such 

. solutions  is  so  similar  to  the  corresponding  applications  of 
equation  (8)  to  gaseous  equilibria  that  no  special  discussion  will 
* be  required  and  the  student  should  have  no  difficulty  in  solving 
the  following  problems,  in  each  of  which  constant  thermodynamic 
environment  is  to  be  assumed. 

6.  Ionization  Equilibria  in  Solution,  (a)  General. — We  have 
already  (XV,  4)  mentioned  the  fact  that  in  solutions  containing 
ions  there  is  every  reason  for  believing  that  the  thermodynamic 
environment  is  a function  of  the  ion  concentration  and  cannot, 

^ therefore,  be  treated  as  a constant,  if  the  ion  concentration 
changes.  For  an  equilibrium  involving  ions  we  are,  therefore, 


256 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


not  in  a position  to  connect  equation  (7)  with  the  mole  fractions 
(in  the  solution)  of  the  molecular  species  involved  in  the  equilib- 
rium, except  by  the  use  of  some  purely  empirical  relationship 
between  the  fugacity  of  a molecular  species,  A,  and  the  ion 
concentration,  C j,  in  the  solution  surrounding  it.  Some  attempts 
to  set  up  such  empirical  relationships  have  been  made  but  they 
have  not  been  very  successful  and  the  problem  of  formulating 
an  equation  of  state  for  solutions  containing  ions  must  await 
the  accumulation  of  the  right  kind  of  experimental  data.  In 
the  following  pages,  therefore,  the  laws  governing  ionization 
equilibria  will  be  presented  as  purely  empirical  relations. 

(b)  The  General  Dilution  Law  for  Uni-univalent  Electro- 
lytes.— The  ionization  of  a uni-univalent  electrolyte,  CA,  is 
expressed  by  the  equation  (XV,  2) 

CA^±C++A~  (18) 


and  the  equilibrium  expression  for  dilute  solutions  is 
[C+HA- 


[CA] 


(see  equation  13) 


If  the  solution  contains  no  other  electrolytes,  then  from  the  defi- 
nition of  a it  follows  (see  problem  10)  that 


[C+][A~]  a2C 


|CA] 


1- 


(19) 


where  C is  the  total  concentration  of  the  electrolyte  in  moles  per 
liter. 


If  we  assume  that  the  ratio, 


Arj 

AoW 


is  a measure  of  the  degree 


of  ionization  of  a uni-univalent  electrolyte  as  expressed  by 
equation  (12,  XVII),  then  the  two  empirical  equations  of  Kraus 
and  Bates  (XVII,  lc)  connecting  conductance  and  concentration 
assume  respectively  the  forms 


= k+k\aC)h  (20) 

1 — a 

and 

logy0-  =k  + k’(aCr  (21) 

I — a 


respectively.  We  have  already  seen  (XVII,  lc  and  Table 
XXII)  that  these  empirical  dilution  laws  are  capable  of  accurately 


Sec.  6] 


CHEMICAL  EQUILIBRIUM 


257 


representing  conductance  data  for  a great  variety  of  electrolytes 
both  in  aqueous  and  in  non-aqueous  solutions.  In  general  the 
sum  of  the  concentrations  of  all  the  species  of  positive  ions  (or  of 
negative  ions)  in  any  solution  is  called  the  total-ion  concentra- 
tion of  that  solution.  In  the  case  now  under  consideration 
(a  solution  of  a uni-univalent  electrolyte)  there  is  evidently  only 
one  species  of  positive  ion,  G-ion,  present  in  significant  amount 
in  the  solution  and  hence  the  total  ion  concentration  of  our  solu- 
tion is  aC,  the  expression  which  occurs  on  the  right-hand  side  of 
equations  (20)  and  (21).  The  facts  represented  by  the  two 
empirical  equations  (20)  and  (21)  may,  therefore,  be  summed  up 
in  the  following  words:  If  we  accept  the  conductance  ratio, 
Arj/Aorio,  as  a measure  of  the  degree  of  ionization  of  a uni- 
univalent electrolyte,  then  the  equilibrium  expression  for  the 
ionization  reaction  of  such  an  electrolyte  is  in  general  not  a 
constant,  at  constant  temperature,  but  increases  as  the  total- 
ion  concentration  of  the  solution  increases. 

(c)  Strong  Electrolytes. — When,  however,  we  come  to  clas- 
sify the  uni-univalent  electrolytes  with  reference  to  their  behavior 
in  this  connection,  we  find  that  for  practical  purposes  they  may 
be  roughly  divided  into  two  groups.  In  the  first  group  fall  the 
strong  electrolytes  (XV,  4 and  XVII,  3).  For  these  electrolytes 
the  term  k'(aC)h  in  equations  (20)  and  (21)  is  always  either  of 
the  same  order  of  magnitude  or  much  larger  than  the  term  k . 
In  other  words  the  total  ion  concentration  is  always  a controlling 
factor  in  the  ionization  of  this  group  of  electrolytes,  the  tendency 
to  ionize,  as  measured  by  the  magnitude  of  the  equilibrium  ex- 
pression, being  greater  the  greater  the  ion  concentration  of  the 
solution. 

(d)  Weak  Electrolytes. — In  the  second  group  fall  the  weak 
electrolytes  (XVII,  3).  For  these  electrolytes  the  term  k'(aC)h 
is  always  small  in  comparison  with  the  term  k and  evidently 
becomes  smaller  the  more  dilute  the  solution.  For  most  pur- 
poses, therefore,  the  term  k'(aC)h  may  be  neglected  in  compari- 
son with  the  term  k in  the  case  of  weak  electrolytes  in  dilute 
solution  and  equations  (20)  and  (21)  on  this  assumption  both 
reduce  to  the  form 

t = const.  = Kc 

1 — a 


(22) 


258  PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


In  other  words  the  ionization  of  weak  electrolytes  in  dilute  solu- 
tion takes  place  in  accordance  with  the  law  of  mass  action. 
(See  equation  19.)  The  weaker  the  electrolyte,  that  is  the 
smaller  the  numerical  value  of  its  ionization  constant,  Kc,  the 
more  closely  is  the  mass  action  law  obeyed.  Even  if  the  ion  con- 
centration of  the  solution  be  greatly  increased  by  adding  a strong 
electrolyte  to  it,  the  resulting  change  in  the  thermodynamic 
environment  does  not  seem  to  very  greatly  influence  the  ioniza- 
tion equilibrium  of  the  weak  electrolyte  and  hence  the  ionization 
constant  of  such  an  electrolyte  may  for  most  purposes  be  assumed 
to  have  approximately  the  same  value  in  all  dilute  solutions  in 
a given  solvent  and  at  a given  temperature,  regardless  of  what 
other  substances  may  be  present,  at  small  concentration,  in  the 
solution.  In  concentrated  solutions  and  for  solutions  containing 
large  additions  of  other  substances,  particularly  electrolytes, 
this  statement  no  longer  holds  true,  however. 

e.  Ionization  in  Mixtures  of  Electrolytes. — In  mixtures  con- 
taining weak  electrolytes  it  will,  as  just  stated,  be  sufficiently 
exact  for  most  purposes  to  assume  that  all  such  electrolytes  obey 
the  law  of  mass  action.  If  the  mixture  contains  only  one  strong 
electrolyte,  the  degree  of  ionization  of  this  electrolyte  may  be 
assumed  to  be  practically  uninfluenced  by  the  presence  of  any 
weak  electrolytes,  but  if  more  than  one  strong  electrolyte  be 
present,  the  ionization  equilibrium  of  each  strong  electrolyte 
will  be  influenced  by  the  presence  of  the  ions  of  the  other  strong 
electrolytes  and  the  general  problem  of  calculating  the  degree  of 
ionization  of  the  strong  electrolytes  in  such  a mixture  is  a complex 
one.  It  is  usually  solved  by  estimating  the  total  ion  concentra- 
tion, Ci,  of  the  mixture  by  a series  of  approximations,  and  then 
assuming  that  the  expression 


(23) 


(24) 


or 


or  some  other  empirical  relationship  holds  for  each  electrolyte  in 
the  mixture,  with  the  same  values  for  the  empirical  constants  as 
are  obtained  from  the  ionization  of  that  same  electrolyte  in  its 
own  pure  solutions  at  the  same  temperature. 


Sec.  6] 


CHEMICAL  EQUILIBRIUM 


259 


This  assumption,  namely,  that  the  equilibrium  expression  for 
a strong  electrolyte  in  a mixture  is  the  same  function  of  the  total 
ion  concentration  as  it  is  in  a pure  solution  of  that  electrolyte, 
was  first  made  by  Arrhenius  and,  is  usually  known  as  the  isohy- 
dric  principle.  The  chief  evidence  in  favor  of  the  validity  of  this 
principle  rests  upon  the  fact  that  the  measured  specific  con- 
ductance of  a mixture  of  two  electrolytes  agrees  fairly  well 
with  the  specific  conductance  calculated  from  this  principle. 
This  agreement  is  likewise  almost  if  not  quite  as  good,  if  the 
isohvdric  principle  be  formulated  in  the  following  approximately 
identical  and  more  convenient  form:  The  equilibrium  expression 
for  a strong  electrolyte  in  a mixture  in  which  the  total  concentration 
of  all  strong  electrolytes  is  C equivalents  per  liter  is  the  same  as  it  is 
in  a pure  solution  of  that  electrolyte  at  the  same  equivalent  concentra- 
tion. The  roughly  approximate  rule  that  all  uni-univalent  strong 
electrolytes  are  ionized  to  the  same  extent  at  the  same  concentra- 
tion has  already  been  stated  (XVII,  3).  In  so  far  as  this  rule  is 
valid,  it  is  evident  that,  if  a solution  contains  several  uni-uni- 
valent strong  electrolytes,  the  equilibrium  expressions  for  all  of 
these  electrolytes  in  that  solution  must  be  equal  to  each  other. 

Problem  17. — On  the  basis  of  the  above  statements  demonstrate  the 
truth  of  the  following  rule:  If  two  uni-univalent  strong  electrolytes  having 
one  ion  in  common  ( e.g .,  KC1  and  NaCl)  are  in  solution  together , the  degree  of 
ionization  of  each  electrolyte  in  the  solution  is  equal  to  the  value  which  that 
electrolyte  has  in  its  own  pure  solution  at  a concentration  equal  to  the  sum  of 
the  concentrations  of  the  two  electrolytes  in  the  mixture. 

Note. — In  all  problems  dealing  with  ionization  equilibria  in  solution  at 
18°  or  25°,  the  necessary  A0  values  are  to  be  computed  from  Table  XXI. 
Other  conductance  data  required,  if  not  to  be  found  in  this  book,  may  be 
taken  from  the  Landolt-Bornstein  tables  or  from  any  other  reliable  source. 
A critical  summary  of  conductance  data  for  aqueous  solutions  of  strong 
electrolytes  has  been  prepared  by  A.  A.  Noyes  and  K.  G.  Falk  [Jour.  Amer. 
Chem  Soc.,  34,  461  to  472  (1912)]. 

Problem  18. — If  0.1  mole  of  HC1  is  added  to  a liter  of  0.1  normal  acetic 
acid  at  18°,  what  will  be  the  acetate  ion  concentration  in  the  resulting  so- 
lution ? 

Problem  19. — How  much  sodium  acetate  must  be  added  to  1 liter  of 
a 0.2  molal  acetic  acid  solution  at  18°  in  order  to  decrease  the  hydrogen  ion 
concentration  100  fold? 

Problem  20. — In  Table  XXVII  are  given  the  values  of  the  conductances 
of  a series  of  aqueous  solutions  of  acetic  acid  at  25°.  Under  V is  given  the 
“dilution”  of  the  solution,  that  is,  the  volume  (in  liters)  containing  one  equiva- 


260 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


lent  of  the  acetic  acid.  Calculate  in  each  instance  the  molal  concentra- 
tion, C,  of  the  solution  and  the  ionization  constant,  Kc,  and  tabulate  your 
results  in  the  corresponding  columns.  The  value  of  A0  for  acetic  acid  at 
25°  can  be  obtained  from  the  data  in  Table  XXI.  For  the  higher  concen- 
trations where  the  viscosity  of  the  solution  is  appreciably  different  from  that 
of  water,  its  influence  may  be  taken  account  of  by  using  equation  (12,  XVII) 
to  calculate  a.  For  this  purpose  the  relative  viscosity,  77/170,  of  each  solution 
is  given  in  the  table.  Finally  as  explained  in  (3b)  and  problem  4,  the  equi- 
librium constant  for  a reaction  in  solution  should  in  general  be  expressed  in 
mole  fractions  instead  of  volume  concentrations.  That  is,  the  quantity  Kc 
could  not  be  expected  to  be  constant  except  in  the  more  dilute  solutions.  In 
general  the  quantity  Kx  should  be  employed.  The  value  of  Kx  for  any 
solution  is  most  conveniently  calculated  from  the  corresponding  value  of  Kc 
with  the  aid  of  equation  (12).'  For  this  calculation  the  density  of  the  solu- 
tion will  be  required  and  density  data  are,  therefore,  included  in  Table 
XXVII.  After  tabulating  the  values  of  Ke  and  Kx  for  all  of  the  solutions, 
construct  graphs  with  values  of  C as  ordinates  and  corresponding  values  of 
Xc  and  Kx  respectively  as  abscissae. 

Table  XXVII 

Conductance  and  Viscosity  Data  for  Acetic  Acid  Solutions  at  25°.  Based 
upon  Measurements  by  Kendall  and  by  Rivett  and  Sidgwick.  For  Illus- 
trating the  Behavior  of  a Typical  Weak  Electrolyte  with  Respect  to  the  Law 
of  Mass  Action. 


V 

K 

C 

R.  & S.. 

I D 25°/25° 

R.  & S. 

L X 104 

A 

i *?/ 

no 

R.  & S. 

OL 

Kc  X 

105 

Kx  X 
107 

6948 . 0 

1.0000 

0.1682 

1 . 0000 

I 

3474.0 

1 . 0000 

0.2496 

1 . 0000 

1737.0 

1 . 0000 

0.3661 

1 . 0000 

868.4 

1 . 0000 

0.5312 

1.0001 

434.2 

1 . 0000 

0.7651 

1 . 0002 

217.1 

1 . 0000 

1.097 

1.0005 

108.6 

1 . 0000 

1.564 

1.001 

54.28 

1.0001 

2.227 

1 . 002 

27.14 



1.0003 

3.165 

1.004 

13.57 



1.0007 

4.485 

1.008 

7.908 

1.0011 





4.618 

1 . 014 

3.954 

1.0022 

3.221 

1 . 028 

1.977 

1.0043 

2.211 

1.056 

(0.7443) 

1.0065 



(13.21) 

1.082 

0.989 

1.0084 

1.443 

1.112 

1.489 

1.0122 

16.71 

1.169 

2.006 

1.0162 

17.89 

1.230 

2 . 977 

1.0235  ! 

18.54 

1.347 

Sec.  7] 


CHEMICAL  EQUILIBRIUM 


261 


Within  what  concentration  range  does  the  ionization  of  acetic  acid  follow 
the  law  of  mass  action?  Up  to  how  high  a concentration  can  the  vis- 
cosity influence  in  the  case  of  acetic  acid  be  neglected,  if  an  accuracy  of  1 per 
cent,  is  desired  in  the  value  of  a?  Up  to  what  concentration  can  volume 
concentrations  instead  of  mole  fractions  be  safely  employed  in  formulating 
the  equilibrium  expression,  without  producing  an  appreciable  deviation 
from  constancy?  The  answers  to  these  questions  are  largely  governed  by 
the  accuracy  of  the  experimental  data  and  the  student  may  form  an  estimate 
of  this  from  his  construction  of  the  above  graphs. 

Problem  21. — On  the  basis  of  the  isohydric  principle  in  its  simpler  form, 
calculate  the  concentration  of  each  species  of  ion  in  the  following  mixtures 
at  18°:  One  liter  of  a solution  contains  (a)  0.1  mole  each  of  KC1  and  NaCl; 
(b)  0.1  mole  each  of  NaN03  and  NaCl;  0.005  mole  of  AgN03  and  0.1  mole 
of  KN03;  0.1  mole  each  of  KN03  and  NaCl;  0.005  mole  of  T1C1  and  0.1 
mole  of  KN03. 

Problem  22. — At  25°  and  0.02 A,  cumic  acid  is  4.88  per  cent,  ionized 
while  glvcollic  acid  is  8.3  per  cent,  ionized.  If  a liter  of  one  solution  is 
mixed  with  a liter  of  the  other,  what  will  be  the  concentration  of  hydrogen 
ion  in  the  resulting  solution? 

Problem  23. — The  same  as  22  with  acetic  acid  in  place  of  cumic  acid. 

Problem  24. — -The  same  as  22  with  hydrochloric  acid  in  place  of  cumic 
acid. 

Problem  25. — -In  what  proportions  should  0.01  normal  cumic  acid  and 
0.015  normal  glycollic  acid  be  mixed  in  order  to  obtain  a solution  (a)  with 
the  minimum  hydrogen-ion  concentration;  (b)  with  the  maximum  hydrogen- 
ion  concentration? 

Problem  26.— Approximately  how  much  potassium  glycollate  should  be 
added  to  one  liter  of  a 0.02  normal  glycollic  acid  solution  in  order  to  give  it 
the  same  hydrogen-ion  concentration  as  a 0.02  normal  cumic  acid  solution? 


B.  Heterogeneous  Systems  at  Constant  Temperature 
7.  The  General  Equilibrium  Law. — Whenever  any  molecular 
species  which  takes  part  in  a chemical  equilibrium  in  a system  at 
.a  constant  temperature  and  under  a constant  external  pressure 
is  represented  in  that  system  by  a pure  crystalline  phase,  then 
the  vapor  pressure  or  fugacity  of  that  molecular  species  cannot  be 
changed  by  any  displacement  of  the  chemical  equilibrium,  for  the 
pressure  of  the  vapor  of  any  substance  in  equilibrium  with  its 
pure  crystals  can  have  only  one  value  at  a given  temperature  and 
pressure.  (Cf.  VI,  1).  In  other  words,  if  the  substance  A is 
involved  in  any  chemical  equilibrium  in  a gas  or  solution,  then  as 
long  as  the  gas  or  solution  is  kept  in  contact  with  and  in  equilib- 


rium with  crystals  of  A,  the  value  of  the  coefficient 


* 

T,  P 


262 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


is  equal  to  zero.  (Cf.  XXII,  3a).  Owing  to  the  very  small 
Vo 

value  of  the  ratio,  , for  crystals  at  moderate  pressures,  it 
Vo 

follows  from  equation  (7,  XII)  that  the  coefficient  (^p) 
must  also  be  extremely  small  under  these  conditions.  Hence  the 
quantity,  dpA=  (^jf)  dx+  which  appears  in  the 

general  law  governing  chemical  equilibrium  (equation  4)  is  either 
zero  or  is  negligibly  small,  if  the  substance  A is  present  in  the 
system  as  a pure  crystalline  phase.  We  thus  reach  the  conclusion 
that  for  a chemical  equilibrium  in  a heterogeneous  system,  not 
only  the  general  thermodynamic  law  governing  the  equilibrium 
but  also  all  the  special  integrated  forms  of  it  (such  as  equations 
7,  8,  10,  11,  and  13)  are  identical  with  the  corresponding  equations 
for  homogeneous  systems,  but  with  the  omission  of  all  terms  refer- 
ring to  substances  present  in  the  system  as  pure  crystalline 
phases.  It  is  clear  that  this  same  statement  would  also  hold 
true  for  any  substances  which  were  present  as  pure  liquids  or 
whose  fugacities  were  kept  constant  by  any  means  whatsoever. 

8.  Crystals  in  Equilibrium  with  Gases. — From  the  general 
conclusion  deduced  in  the  preceding  section  it  is  evident  that  for 
the  equilibrium  represented  by  the  equation  (cf.  XIX,  1) 

NH4Cl  = NH~3  + HCj  (25) 

in  which  crystalline  ammonium  chloride  is  in  equilibrium  with 
its  gaseous  dissociation  products,  the  equilibrium  law  would  be 
Pnhs'Phci  = const.  = Kp  (26) 

and 


Cnh3'Chci  — const.  — Kc  (27) 

corresponding  to  the  general  equations  (7)  and  (8). 

Problem  28.  -Formulate  the  equilibrium  laws  of  the  following  reactions: 


CaC03  = CaO  + C02 

(28) 

3Fe+4H20  = Fe304-f-4H2 

(29) 

BaC03aq  + SrS04  = BaS04+SrC03aq 

(30) 

PbO  + NH4CI  = Pb(OH)Cl  + NH3 

(31) 

NH40C0NH2  = 2NH3+C02 

(32) 

C + C02  = 2C0 

(33) 

Fe203  + 3C0  = 2Fe  + 3C02 

(34) 

k 


Sec.  9] 


CHEMICAL  EQUILIBRIUM 


263 


The  equilibrium  constants  for  the  above  reactions  have  all 
been  experimentally  determined.  The  principle  features  of  an 
apparatus  for  determining  the  value  of  the  equilibrium  constant 
for  reaction  (29)  are  illustrated  in  Fig.  34.  The  air  is  first  com- 
pletely pumped  out  of  the  apparatus  and  the  granular  mixture  of 
solid  Fe  + Fe304  in  the  horizontal  reaction  tube  is  heated 
electrically  to  the  desired  temperature.  After  equilibrium  is 
attained  the  total  pressure  inside  the  apparatus  is  registered  by 
the  manometer.  This  pressure  is  equal  the  partial  pressure  of 
the  hydrogen  produced  by  the  reaction  plus  the  partial  pressure 
of  the  water  vapor.  This  latter  pressure  is  kept  constant  owing 
to  the  presence  in  the  apparatus  of  liquid  water  at  a definite 
temperature  which  can  be  controlled  by  means  of  a constant  tem- 
perature bath  into  which  the  whole  apparatus  is  plunged. 


Fig.  34. — Illustrating  the  principal  features  of  an  apparatus  for  determin- 
ing the  equilibrium  constant  for  the  reaction  3Fe  + 4H20  = FeAb  + 4H2. 
T indicates  thermocouples  for  measuring  the  temperature  inside  the  reac- 
tion tube.  Experiments  with  this  apparatus  are  described  by  G.  Preuner, 
Z.  physik.  Chem.  47,  392  (1904). 

Problem  29. — In  one  experiment  with  the  apparatus  shown  in  Fig.  34, 
the  constant  temperature  bath  surrounding  the  whole  apparatus  was  main- 
tained at  38°  and  the  reaction  tube  at  900°.  When  equilibrium  had  been 
reached  the  manometer  registered  121  mm.  The  vapor  pressure  of  water  at 
38°  is  49.7  mm.  Calculate  the  equilibrium  constant  for  reaction  (29)  at 
900°.  If  the  constant  temperature  bath  had  been  regulated  at  20°  instead 
of  38°  in  the  above  experiment  what  would  the  manometer  have  read  after 
the  attainment  of  equilibrium?  (See  IV,  1,  Table  IX.) 

(The  mean  value  obtained  experimentally  under  these  conditions  was 
43.7  mm.) 


264 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


9.  Crystals  in  Equilibrium  with  Solutions. — On  the  basis  of 
the  reasoning  given  in  section  7 above,  the  student  should  have  no 
difficulty  in  formulating  the  equilibrium  expression  for  any  het- 
erogeneous chemical  equilibrium  in  a solution  of  constant  thermo- 
dynamic environment. 

Problem  30. — When  phenanthrene  picrate  is  dissolved  in  alcohol  it  dis- 
sociates partially  into  free  phenanthrene  and  free  picric  acid.  Assuming  that 
the  thermodynamic  environment  within  the  solution  remains  constant  for- 
mulate the  equilibrium  law  for  the  above  reaction  when  the  solution  is  kept 
saturated  (1)  with  picric  acid;  (2)  with  phenanthrene;  (3)  with  the  solid  salt; 
(4)  with  both  the  solid  salt  and  the  acid. 

Aqueous  solutions  saturated  with  crystalline  electrolytes  form 
an  important  group  of  heterogeneous  equilibria  in  solutions 
whose  thermodynamic  environment  is  a function  of  the  ion  con- 
centration. Take  for  example  the  equilibrium  represented  by  the 
equation 

CA  — C++A-  (35) 

where  a solution  is  saturated  with  the  solid  uni-univalent  elec- 
trolyte, CA.  As  explained  in  section  7 above,  the  general  thermo- 
dynamic law  for  this  equilibrium  (equation  7)  becomes 

Pc+‘Pa~  = const.  (36) 

where  p represents  the  fugacity  of  the  ion  species  indicated. 

In  order  to  obtain  a relation  involving  the  concentrations  of 
the  two  ion  species  in  the  solution,  it  is  necessary  to  know  the 
relation  between  fugacity  and  concentration  for  each  ion  species. 
(Cf.  XXII,  6a.)  Since  the  law  of  mass  action  does  not  hold 
(XXII,  6b  and  c)  for  the  ionization  equilibrium  of  a strong  elec- 
trolyte, it  follows  that  fugacity  and  concentration  cannot  be 
proportional  to  each  other  for  all  of  the  molecular  and  ion  species 
involved  in  such  an  equilibrium,  because  we  have  already  seen 
(XXII,  3b)  that  such  a proportionality  would  make  the  mass 
action  law  a thermodynamic  necessity.  In  the  equilibrium 
expression  for  the  ionization  of  a strong  electrolyte  in  a homo- 
geneous system,  however,  the  un-ionized  molecules  are  involved 
as  well  as  the  ions,  while  if  the  solution  is  kept  saturated  with 
crystals  of  the  electrolyte,  the  fugacity  of  the  un-ionized  mole- 
cules is  thereby  kept  constant  and  hence  drops  out  of  the  expres- 
sion, leaving  simply  equation  (36). 


Sec.  9] 


CHEMICAL  EQUILIBRIUM 


265 

The  failure  of  the  mass  action  law  in  the  case  of  strong  electro- 
lytes might,  therefore,  conceivably  be  very  largely  due  to  a rapid 
variation  of  the  fugacity  of  the  un-ionized  molecules  with  the 
ion  concentration  of  the  solution  and  it  might  still  be  possible  in 
dilute  solutions  that  the  fugacity  of  an  ion  species  was  not  far 
from  being  proportional  to  its  concentration.  In  other  words 
the  change  of  thermodynamic  environment  brought  about  by  a 
change  in  the  ion  concentration  of  a solution  might  have  a very 
large  influence  upon  the  fugacity  of  the  un-ionized  molecules  of 
strong  electrolytes  and  only  a small  effect  upon  the  fugacity  of  an 
ion  species.  If  we  assume,  therefore,  that  in  sufficiently  dilute 
solutions  the  fugacity  of  an  ion  species  is  proportional  to  its 
concentration,  equation  (36)  would  evidently  become 

[C+]  [A-]  = const.  = (ao$0)2  (37) 

where  S0  is  the  solubility  of  the  salt  in  pure  water  and  a0  its 
degree  of  ionization  in  this  saturated  solution.  The  validity  of 
equation  (37)  can  be  tested  by  experiment.  Suppose  we  have  a 
saturated  solution  of  the  slightly  soluble  salt,  T1C1.  By  addi- 
tions of  KC1  and  of  T1N03  respectively  to  this  solution  we  can 
vary  the  values  of  [T1+]  and  [Cl~]  in  this  solution  through  wide 
limits  and  can  thus  discover  whether  or  not  their  product  re- 
mains constant  as  required  by  equation  (37). 

Experiments  of  this  character  have  been  carried  out  with  a 
number  of  electrolytes  and  the  conclusions  reached  may  be 
summed  up  as  follows:  In  sufficiently  dilute  solutions  saturated 
with  a slightly  soluble  uni-univalent  salt  the  product  of  the  concen- 
trations of  the  ion  species  of  that  salt  is  constant.  This  statement 
'which  is  expressed  mathematically  by  equation  (37)  is  usually 
known  as  the  solubility -product  law.  It  is  more  nearly  true  the 
less  soluble  the  salt  and  the  more  dilute  the  solution.  For  most 
purposes  it  holds  sufficiently  well  for  all  cases  where  the  solu- 
bility of  the  salt  and  the  ion  concentration  of  the  solution  are 
both  not  more  than  a few  hundredths  of  an  equivalent  per  liter. 
An  example  illustrating  the  accuracy  with  which  the  principle 
holds  for  a specific  case  is  shown  in  Fig.  35.  This  figure  is  based 
upon  measurements  made  by  Bray  and  Winninghof  with  solu- 
tions of  thallous  chloride  at  25°.  The  solubility  of  this  salt  in 
pure  water  at  25°  is  0.01629  equivalents  per  liter.  Its  solubility 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


266 

in  solutions  of  T1N03,  KC1  and  KN03  of  varying  strengths  was 
determined  and  the  values  of  the  solubility  product  [Tl+]  [Cl-]  in 
these  different  solutions  were  calculated  with  the  aid  of  the  iso- 
hydric  principle  and  are  plotted  as  ordinates  in  the  figure,  corre- 
sponding values  of  the  total-ion  concentration  of  the  solution, 
raised  to  the  \ power  (CV),  being  plotted  as  abscissae. 


Fig.  35. — Illustrating  the  variation  of  the  solubility  product  [Tl+]  [Cl-] 
in  solutions  saturated  with  thallous  chloride  and  containing  respectively 
the  salts  T1N03,  KC1,  and  KN03  at  different  concentrations.  The  point 
marked  by  the  cross  represents  the  value  for  a pure  aqueous  solution  of 
thallous  chloride.  The  total  ion-concentration  for  each  solution  is  calcu- 
lated by  means  of  the  isohydric  principle.  This  figure  is  based  upon  ex- 
perimental data  obtained  by  Bray  and  Winninghof  at  25°. 

It  is  evident  from  this  figure  that  the  solubility  product  for 
this  salt  is  not  constant,  but  increases  regularly  as  the  total-ion 
concentration  of  the  solution  increases,  being  in  fact  approxi- 
mately a linear  function  of  C{ \ That  is,  the  solubility  product 
law  could  be  more  accurately  expressed  by  an  equation  of  the 
form 

[C+]  [A-]  = k + k'C?  (38) 

where  h is  approximately  J and  the  constants  Jc  and  k'  depend 
upon  the  nature  of  the  ion  species  present  in  the  solution.  This 
equation  has  the  same  form  as  the  Kraus  equation,  equation  (20), 
for  the  ionization  of  strong  electrolytes.  The  straight  lines 


Sec.  9] 


CHEMICAL  EQUILIBRIUM 


267 


drawn  in  Fig.  35  are  graphs  of  the  equations  indicated  on  the 
figure.  Although  the  solubility  product  very  evidently  increases 
as  the  total-ion  concentration  increases,  the  change  is  not  very 
rapid  and  for  ion  concentrations  not  greater  than  0.04  ( i.e 
Ci  = 0.2)  the  solubility  product  for  thallous  chloride  may 
evidently  be  taken  as  constant  within  10  per  cent.  For  less 
soluble  salts  the  magnitude  of  the  solubility  product  and  its  rate 
of  change  with  the  ion  concentration  are  both  much  smaller. 

The  application  of  the  solubility  product  law  to  salts  of  higher 
valence  types  (XV,  3b)  or  to  salts  of  any  valence  type  when  the 
solution  contains  a salt  of  a higher  valence  type  is  complicated 
by  the  presence  of  intermediate  ions  (XVII,  3)  whose  concentra- 
tions and  equilibrium  relations  are  not  known.  Thus,  for 
example,  the  solubility  product  law  for  a solution  saturated  with 
PbCl2  would  read 

[Pb++]  [Cl-]2  = const.  (39) 

and  if  either  the  chloride-ion  concentration  or  the  lead-ion  con- 
centration of  the  solution  were  increased  by  the  addition  of  a salt 
with  the  common  ion,  we  should  expect  the  solubility  of  the 
PbCl2  to  be  diminished.  This  is  what  actually  happens  when  a 
chloride,  such  as  KC1,  is  added  to  a saturated  solution  ofPbCl2. 
When,  however,  a salt  having  the  common  bivalent  ion  (Pb- 
(N03)2,  for  example)  is  added  the  solubility  of  the  PbCl2  may 
actually  be  increased.  The  addition  of  the  Pb(N03)2  doubtless 
increases  the  lead-ion  concentration  of  the  solution  somewhat 
but  at  the  same  time  it  decreases  the  chloride  ion  concentration 
owing  to  the  occurrence  of  the  reaction 

Pb++  + Cl-  = PbCl+  (40) 

resulting  in  the  formation  of  the  intermediate  ion,  PbCl+.  This 
decrease  in  the  chloride-ion  concentration  may  more  than  com- 
pensate for  the  increase  in  the  lead-ion  concentration  and  can 
thus  produce  an  actual  increase  in  the  solubility  instead  of  a 
decrease.  A general  discussion  of  solubility  relations  when  salts 
of  higher  valence  types  are  involved  can  be  found  in  papers  by 
A.  A.  Noyes,  Bray  and  Harkins. 

Problem  31. — When  conductivity  water  having  a specific  conductance  of 
0.70- 10~6,  reciprocal  ohms  at  18°  is  saturated  with  silver  iodate  at  this 


268 


PRINCIPLES  OF  PHYSICAL  CHEMISTRY  [Chap.  XXII 


temperature,  the  specific  conductance  of  the  saturated  solution  is  found  to 
be  12.6  X 10-6  reciprocal  ohms.  Calculate  the  solubility  product  (a0S0)2, 
for  AgI03  at  18°. 

Problem  31  shows  that  with  the  aid  of  conductance  measure- 
ments it  is  possible  to  determine  the  solubility  product  ( a0S0 )2 
without  knowing  either  the  solubility  (So)  itself  or  the  degree  of 
ionization  (a0)  in  the  saturated  solution. 

Problem  32.— From  the  value  of  a0S0  obtained  in  problem  31  compute  the 
solubility  of  silver  iodate  at  18°,  first  under  the  assumption  that  its  degree 
of  ionization  in  the  saturated  solution  is  the  same  as  that  of  potassium  iodate 
at  the  same  concentration  and  second  under  the  assumption  that  it  is  the 
same  as  that  of  silver  nitrate  at  the  same  concentration. 

Problem  33. — From  the  results  of  the  preceding  problems  calculate  the 
solubility,  S,  of  AgI03  (a)  in  a 0.01  molal  solution  of  AgN03  and  (b)  in  a 
0.02  molal  solution  of  KI03  at  18°. 

Problem  34. — From  the  results  of  the  preceding  problems  calculate  the 
amount  of  AgI03  which  will  dissolve  in  a 0.02  formal  solution  of  NH3  at  18° 
assuming  that  the  complex-ion  Ag(NH3)2+,  forms  quantitatively  (Cf. 
XV,  problem  5 and  XVIII,  problem  13).  * 

Problem  35. — The  solubility  of  silver  chloride  in  water  at  25°  is  2 mg.  per 
liter.  If  a precipitate  of  AgCl  is  washed  with  two  liters  of  wash  water  at  25°, 
how  much  AgN03  should  the  wash  water  contain  in  order  to  prevent  the 
dissolving  of  more  than  0.05  mg.  of  AgCl  during  the  operation  of  washing? 
Assume  equilibrium  to  be  established  during  the  washing. 

Problem  36.— The  solubility  of  thallous  chloride  is  0.0163  and  that  of 
thallous  sulpho-cyanate  0.0149  mole  per  liter.  How  many  grams  of  thal- 
lium can  be  obtained  from  one  liter  of  a solution  which  is  saturated  with 
both  salts  at  the  same  time? 

Problem  37. — State  and  interpret  in  the  light  of  the  facts  presented  in 
this  chapter  the  effect  on  the  solubility  of  cuprous  iodide  (Cul)  produced  by 
the  presence  of  the  following  substances  in  dilute  aqueous  solution:  (1) 
Cuprous  chloride,  (2)  iodine,  (3)  cupric  sulphate,  (4)  NH3,  (5)  lithium  ni- 
trate, (6)  a strong  oxidizing  agent,  (7)  sugar,  (8)  potassium  iodide,  (9)  hy- 
drochloric acid,  (10)  acetic  acid.  State  also,  as  far  as  possible,  the  relative 
magnitudes  of  the  effects  of  the  above  substances. 

Problem  38.— What  would  be  the  direction  and  relative  magnitude 
(whether  large  or  small)  of  the  effects  on  the  solubility  of  silver  benzoate 
(So  = 0.014  mol.  per  liter)  produced  by  adding  separately  to  one  liter  of  a 
saturated  solution,  0.05  mol.  of  each  of  the  following  substances?  Explain 
what  occurs  in  each  case.  AgC103,  CeHsCOOLi,  LiN03,  NH3,  LiCN, 
HN03,  HC1.  Benzoic  acid  is  a weak  acid. 


